Rotational dynamics | PhysicsRank

A door and a compact wheel can have the same mass yet respond very differently to a turning effect. Rotation cares about where the mass sits relative to the axis. We will make that extra piece of the story explicit.

We build fixed-axis rotation from particle motion, derive useful inertia relations and test their geometric limits. Radius of gyration and the perpendicular-axis theorem then become consequences of mass distribution rather than isolated formulas.

What you will learn to do

Connect angular motion to point speeds and accelerations.
Calculate signed torques and distinguish force balance from torque balance.
Derive moment of inertia and use the correct axis.
Interpret radius of gyration as a mass-weighted rms distance.
Derive and restrict the perpendicular-axis theorem to planar distributions.
Solve coupled translation and fixed-axis rotation and check rotational work.

Useful before this chapterThe cross product: perpendicular effects and direction →Forces that turn a moving object →Work, stored energy and power →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Torque, rotational inertia and fixed-axis dynamics
Learning goals:
- Connect angular motion to point speeds and accelerations.
- Calculate signed torques and distinguish force balance from torque balance.
- Derive moment of inertia and use the correct axis.
- Interpret radius of gyration as a mass-weighted rms distance.
- Derive and restrict the perpendicular-axis theorem to planar distributions.
- Solve coupled translation and fixed-axis rotation and check rotational work.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

BEFORE YOU READ

Bring back one idea

Try a sentence or a sketch before opening the comparison. If this is new, use the explanation to get started.

What two geometric choices are needed before calculating torque?

Compare your reasoning

Choose the axis or origin and the force’s line of action. The perpendicular distance between them sets the lever arm. A force magnitude alone cannot determine torque.

Revisit the explanation →

01 / Torque, rotational inertia and fixed-axis dynamics

Rigid rotation gives every point the same angular change

A rigid body keeps fixed distances between its material points. When it rotates about a fixed axis, every point undergoes the same angular displacement in the same time, even though points farther from the axis travel longer arcs. Angular position θ is measured in radians; angular velocity ω = dθ/dt and angular acceleration α = dω/dt describe its rate and rate of change.

For a point at perpendicular distance r from the axis, arc displacement is rθ, tangential speed is r|ω| and tangential acceleration has magnitude r|α|. There is also radial acceleration rω² toward the axis. Equal angular acceleration does not mean equal linear acceleration at different radii.

For constant α, the familiar constant-acceleration equations have angular counterparts. Their use requires the same constant-rate assumption. A changing torque, changing moment of inertia or speed-dependent resistance can make α variable, requiring derivatives or integrals instead.

\omega=\omega_0+\alpha t,\qquad\Delta\theta=\omega_0t+\frac12\alpha t^2,\qquad\omega^2-\omega_0^2=2\alpha\Delta\theta

These relations apply to rotation about a fixed axis with constant angular acceleration. Angles are in radians.

WORKED EXAMPLE

The same rotation produces different point speeds

A wheel starts from rest and has constant angular acceleration 3 rad/s² for 2 s. Find its angular speed, angle turned and the tangential speed of a point 0.40 m from the axis.

Choose an approach

Find the shared angular quantities first, then multiply angular speed by the point’s radius.

Integrate the constant angular acceleration over time.
$\omega=3(2)=6\ \mathrm{rad/s}$
The angle is the area under the angular-velocity graph.
$\Delta\theta=\frac12(3)(2^2)=6\ \mathrm{rad}$
Use the local radius to find tangential speed.
$v=r\omega=0.40(6)=2.4\ \mathrm{m/s}$

What this gives usThe wheel turns 6 rad and reaches 6 rad/s. The selected point moves tangentially at 2.4 m/s. A point at half the radius has half that speed, despite the same angular velocity.

Check the result another way

The triangular ω–t area is ½ × 2 × 6 = 6 rad. At the final instant the selected point also has inward acceleration rω² = 14.4 m/s², separate from its tangential acceleration 1.2 m/s².

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Worked-example topic: The same rotation produces different point speeds
Problem: A wheel starts from rest and has constant angular acceleration 3 rad/s² for 2 s. Find its angular speed, angle turned and the tangential speed of a point 0.40 m from the axis.

Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Torque, rotational inertia and fixed-axis dynamics
Focus: Rigid rotation gives every point the same angular change
Learning goals:
- Connect angular motion to point speeds and accelerations.
- Calculate signed torques and distinguish force balance from torque balance.
- Derive moment of inertia and use the correct axis.
- Interpret radius of gyration as a mass-weighted rms distance.
- Derive and restrict the perpendicular-axis theorem to planar distributions.
- Solve coupled translation and fixed-axis rotation and check rotational work.
Equations in this section (LaTeX):
\omega=\omega_0+\alpha t,\qquad\Delta\theta=\omega_0t+\frac12\alpha t^2,\qquad\omega^2-\omega_0^2=2\alpha\Delta\theta
Meaning: These relations apply to rotation about a fixed axis with constant angular acceleration. Angles are in radians.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

02 / Torque, rotational inertia and fixed-axis dynamics

Torque is the turning component of a force about a chosen axis

The torque of a force about a point is r × F. For rotation about a fixed axis, use the component of that vector along the axis. In a planar diagram, it is often simplest to use force times perpendicular lever arm with a sign for clockwise or anticlockwise tendency.

A force can be large yet have zero torque about an axis if its line of action passes through that axis. A hinge force applied on the rotation axis is an example. This is why taking moments about a pivot can eliminate an unknown support force from a rotational equation without pretending that the force does not exist.

Choose the axis before inserting distances. The radius to the application point is not always the perpendicular lever arm. A force applied at an angle contributes only its component tangent to the circular path about that axis. Radial components can affect support or stress without contributing torque about the axis.

\tau_z=(\mathbf r\times\mathbf F)_z=rF\sin\phi=F d_\perp

The angle φ is between the position vector r and force F. Assign a sign to the chosen axis component.

WORKED EXAMPLE

A short lever can outweigh a longer one through force direction

About a pivot, a 10 N force acts at radius 0.30 m with sin φ = 0.6 and tends anticlockwise. A second force of 4 N acts tangentially at radius 0.20 m and tends clockwise. Find the net torque, taking anticlockwise as positive.

Choose an approach

Calculate signed moments separately using each perpendicular component.

The first torque is positive but not the full rF.
$\tau_1=0.30(10)(0.6)=1.8\ \mathrm{N\,m}$
The tangential second force has its full lever arm.
$\tau_2=-0.20(4)=-0.8\ \mathrm{N\,m}$
Add the signed contributions.
$\tau_{\mathrm{net}}=1.8-0.8=1.0\ \mathrm{N\,m}$

What this gives usThe resultant torque is 1.0 N·m anticlockwise. A torque sign indicates the tendency to change angular momentum about the chosen axis.

Check the result another way

The first force’s perpendicular lever arm is 0.30 × 0.6 = 0.18 m. Multiplying 10 N by that distance reproduces its 1.8 N·m moment.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Worked-example topic: A short lever can outweigh a longer one through force direction
Problem: About a pivot, a 10 N force acts at radius 0.30 m with sin φ = 0.6 and tends anticlockwise. A second force of 4 N acts tangentially at radius 0.20 m and tends clockwise. Find the net torque, taking anticlockwise as positive.

Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Torque, rotational inertia and fixed-axis dynamics
Focus: Torque is the turning component of a force about a chosen axis
Learning goals:
- Connect angular motion to point speeds and accelerations.
- Calculate signed torques and distinguish force balance from torque balance.
- Derive moment of inertia and use the correct axis.
- Interpret radius of gyration as a mass-weighted rms distance.
- Derive and restrict the perpendicular-axis theorem to planar distributions.
- Solve coupled translation and fixed-axis rotation and check rotational work.
Equations in this section (LaTeX):
\tau_z=(\mathbf r\times\mathbf F)_z=rF\sin\phi=F d_\perp
Meaning: The angle φ is between the position vector r and force F. Assign a sign to the chosen axis component.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

03 / Torque, rotational inertia and fixed-axis dynamics

Zero resultant force does not imply zero resultant torque

Two equal and opposite forces can act along different parallel lines. Their net force is zero, but their turning effects can add. Such a pair is a couple. Its torque magnitude is Fd, where d is the perpendicular separation of the lines of action.

A couple’s torque is independent of the origin. Shifting the origin changes the total torque by a term involving the displacement of the origin crossed with the net force. For a couple the net force is zero, so that correction vanishes. A single force generally does not have this origin independence.

For a rigid body in static equilibrium, both the net force and the net torque must vanish. Either condition alone is insufficient. Two balancing forces can still form a couple; several torques can cancel while a nonzero force translates the body. Count translational and rotational equilibrium conditions separately.

WORKED EXAMPLE

A pair that turns without translating the centre of mass

Two horizontal forces of magnitude 2 N act on a rigid body: the upper force points left at y = +0.25 m and the lower force points right at y = −0.25 m. Find the resultant force and torque about the origin, with x right and y up.

Choose an approach

The forces cancel as vectors, but their moments about the origin have the same sign.

Add the horizontal force components.
$F_{\mathrm{net},x}=-2+2=0$
Use τz = xFy − yFx for the two horizontal forces.
$\tau_z=-0.25(-2)-(-0.25)(2)=1.0\ \mathrm{N\,m}$

What this gives usThe net force is zero and the couple torque is 1.0 N·m out of the page, corresponding to anticlockwise tendency.

Check the result another way

The force-line separation is 0.50 m, so Fd = 2 × 0.50 = 1.0 N·m. The same value is obtained about a point on either force line.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Worked-example topic: A pair that turns without translating the centre of mass
Problem: Two horizontal forces of magnitude 2 N act on a rigid body: the upper force points left at y = +0.25 m and the lower force points right at y = −0.25 m. Find the resultant force and torque about the origin, with x right and y up.

Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

STOP & THINK

A rigid body has zero net force but a nonzero net torque. Is it in static equilibrium?

YesNoOnly if the forces have equal magnitudes

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Torque, rotational inertia and fixed-axis dynamics / Zero resultant force does not imply zero resultant torque
Problem: A rigid body has zero net force but a nonzero net torque. Is it in static equilibrium?
Choices:
1. Yes
2. No
3. Only if the forces have equal magnitudes

Help me choose a model and explain my reasoning before checking my answer.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Torque, rotational inertia and fixed-axis dynamics
Focus: Zero resultant force does not imply zero resultant torque
Learning goals:
- Connect angular motion to point speeds and accelerations.
- Calculate signed torques and distinguish force balance from torque balance.
- Derive moment of inertia and use the correct axis.
- Interpret radius of gyration as a mass-weighted rms distance.
- Derive and restrict the perpendicular-axis theorem to planar distributions.
- Solve coupled translation and fixed-axis rotation and check rotational work.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

04 / Torque, rotational inertia and fixed-axis dynamics

Moment of inertia weights distance from the axis twice

A point mass at distance r from a rotation axis has speed rω, so its kinetic energy is ½mr²ω². Sum over the points of a rigid body: the common ω² factors out and leaves I = Σmr². This quantity is the moment of inertia about that axis.

The perpendicular distance to the axis is squared. Mass far from the axis contributes disproportionately. Two bodies with the same mass and outer radius can therefore respond differently to the same torque if their mass distributions differ. The axis is part of the specification of I; a body does not have one universal moment of inertia.

For a continuous distribution, replace the sum by ∫r² dm. Choose a mass element compatible with the geometry and density. A uniform rod can be divided into length elements, a disc into rings, and a lamina into area elements. The radius in the integral is always the perpendicular distance to the chosen axis.

I=\sum_i m_ir_i^2=\int r^2\,dm,\qquad K_{\mathrm{rot}}=\frac12I\omega^2

Moment of inertia has units kg·m². Its radius is a distance to an axis, not necessarily a distance from a body’s centre.

WORKED EXAMPLE

Redistributing mass changes rotational resistance

Two small masses, 1 kg and 3 kg, are fixed to a light rod at perpendicular distances 0.40 m and 0.20 m from a rotation axis. Find I and the kinetic energy at ω = 5 rad/s. Neglect each mass’s size and the rod’s mass.

Choose an approach

Add each mass times its own squared radius, then use the common angular speed.

The two contributions are 0.16 and 0.12 kg·m².
$I=1(0.40)^2+3(0.20)^2=0.28\ \mathrm{kg\,m^2}$
Calculate rotational kinetic energy.
$K=\frac12(0.28)(5^2)=3.5\ \mathrm J$

What this gives usThe moment of inertia is 0.28 kg·m² and the rotational kinetic energy is 3.5 J. The smaller mass contributes more to I because it is farther from the axis.

Check the result another way

The point speeds are 2 m/s and 1 m/s. Their individual kinetic energies sum to ½(1)(4) + ½(3)(1) = 3.5 J, independently checking the rotational expression.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Worked-example topic: Redistributing mass changes rotational resistance
Problem: Two small masses, 1 kg and 3 kg, are fixed to a light rod at perpendicular distances 0.40 m and 0.20 m from a rotation axis. Find I and the kinetic energy at ω = 5 rad/s. Neglect each mass’s size and the rod’s mass.

Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Torque, rotational inertia and fixed-axis dynamics
Focus: Moment of inertia weights distance from the axis twice
Learning goals:
- Connect angular motion to point speeds and accelerations.
- Calculate signed torques and distinguish force balance from torque balance.
- Derive moment of inertia and use the correct axis.
- Interpret radius of gyration as a mass-weighted rms distance.
- Derive and restrict the perpendicular-axis theorem to planar distributions.
- Solve coupled translation and fixed-axis rotation and check rotational work.
Equations in this section (LaTeX):
I=\sum_i m_ir_i^2=\int r^2\,dm,\qquad K_{\mathrm{rot}}=\frac12I\omega^2
Meaning: Moment of inertia has units kg·m². Its radius is a distance to an axis, not necessarily a distance from a body’s centre.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

05 / Torque, rotational inertia and fixed-axis dynamics

Derive a familiar inertia once instead of treating it as a label

For a uniform thin rod of length L and mass M, take the axis through its centre perpendicular to its length. Let x measure position along the rod. Uniform linear density gives dm = (M/L)dx, and the perpendicular distance to the axis is |x|.

Integrate x² from −L/2 to +L/2. The square means equal pieces on opposite sides contribute equally. The result ML²/12 depends on the rod being uniform and thin and on the stated central perpendicular axis. A rod rotating about its length has a different, much smaller inertia that depends on its finite cross-section.

To shift an axis parallel to itself, use I = Icom + Md², where d is the perpendicular separation from the parallel axis through the centre of mass. The reference axis must pass through the centre of mass for this simple formula. Adding Md² between two arbitrary parallel axes can be wrong.

I_{\mathrm{rod,centre}}=\frac ML\int_{-L/2}^{L/2}x^2\,dx=\frac{ML^2}{12},\qquad I_{\mathrm{parallel}}=I_{\mathrm{com}}+Md^2

The shift term is nonnegative, so among parallel axes the centre-of-mass axis has the least inertia.

WORKED EXAMPLE

Move a rod’s rotation axis from its centre to an end

A uniform thin rod has mass 3 kg and length 2 m. Find its moment of inertia about an axis through one end perpendicular to the rod.

Choose an approach

Start with the central parallel axis and shift it by L/2.

The central inertia follows from the uniform-rod integral.
$I_{\mathrm{com}}=\frac{3(2^2)}{12}=1\ \mathrm{kg\,m^2}$
The separation of the axes is 1 m.
$I_{\mathrm{end}}=1+3(1^2)=4\ \mathrm{kg\,m^2}$

What this gives usThe end-axis inertia is 4 kg·m², four times the central value. The same rod is harder to angularly accelerate about its end under the same torque.

Check the result another way

Direct integration from x = 0 to x = 2 m gives (3/2)∫x² dx = (3/2)(8/3) = 4 kg·m².

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Worked-example topic: Move a rod’s rotation axis from its centre to an end
Problem: A uniform thin rod has mass 3 kg and length 2 m. Find its moment of inertia about an axis through one end perpendicular to the rod.

Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Torque, rotational inertia and fixed-axis dynamics
Focus: Derive a familiar inertia once instead of treating it as a label
Learning goals:
- Connect angular motion to point speeds and accelerations.
- Calculate signed torques and distinguish force balance from torque balance.
- Derive moment of inertia and use the correct axis.
- Interpret radius of gyration as a mass-weighted rms distance.
- Derive and restrict the perpendicular-axis theorem to planar distributions.
- Solve coupled translation and fixed-axis rotation and check rotational work.
Equations in this section (LaTeX):
I_{\mathrm{rod,centre}}=\frac ML\int_{-L/2}^{L/2}x^2\,dx=\frac{ML^2}{12},\qquad I_{\mathrm{parallel}}=I_{\mathrm{com}}+Md^2
Meaning: The shift term is nonnegative, so among parallel axes the centre-of-mass axis has the least inertia.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

06 / Torque, rotational inertia and fixed-axis dynamics

Radius of gyration compresses a mass distribution into one distance

The radius of gyration k about an axis is defined by I = Mk². It is the distance at which the entire mass could be concentrated in an ideal thin ring about that axis to produce the same moment of inertia. It is an equivalent distribution parameter, not necessarily a physical surface or cavity in the body.

The definition gives k² = (1/M)∫r² dm: a mass-weighted mean squared distance. Consequently k is a root-mean-square distance, not the ordinary average radius. Squaring gives greater weight to distant mass, so the two averages usually differ.

Changing the axis changes k even if the body is unchanged. A parallel-axis shift gives k² = kcom² + d². For composite bodies, add their inertias about the same axis first and divide by the total mass. Averaging the individual radii of gyration directly generally does not give the composite value.

WORKED EXAMPLE

Equal mass and radius do not imply equal rotational inertia

A thin hoop and a uniform thin disc each have mass M and outer radius R, and rotate about their central symmetry axes. The hoop has I = MR²; the disc has I = ½MR². Find each radius of gyration and compare their angular accelerations under the same net torque.

Choose an approach

Use k = √(I/M). Then use the fixed-axis relation α = τ/I.

The hoop’s mass all lies at radius R.
$k_{\mathrm{hoop}}=R$
The disc includes mass at smaller radii.
$k_{\mathrm{disc}}=\frac{R}{\sqrt2}$
At equal torque, the smaller inertia gives the larger acceleration.
$\frac{\alpha_{\mathrm{disc}}}{\alpha_{\mathrm{hoop}}}=\frac{I_{\mathrm{hoop}}}{I_{\mathrm{disc}}}=2$

What this gives usThe disc’s radius of gyration is R/√2 and it has twice the angular acceleration of the hoop under the same torque. Neither conclusion requires a different outer radius.

Check the result another way

The disc’s inertia can be checked by rings: dm = (2M/R²)r dr gives I = (2M/R²)∫₀ᴿr³dr = MR²/2.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Worked-example topic: Equal mass and radius do not imply equal rotational inertia
Problem: A thin hoop and a uniform thin disc each have mass M and outer radius R, and rotate about their central symmetry axes. The hoop has I = MR²; the disc has I = ½MR². Find each radius of gyration and compare their angular accelerations under the same net torque.

Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

STOP & THINK

A body has mass 4 kg and moment of inertia 1 kg·m² about a stated axis. What is its radius of gyration?

Your answer (m)Use a decimal, a fraction such as 1/3, or scientific notation.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Torque, rotational inertia and fixed-axis dynamics / Radius of gyration compresses a mass distribution into one distance
Problem: A body has mass 4 kg and moment of inertia 1 kg·m² about a stated axis. What is its radius of gyration?
Requested answer unit: m

Help me choose a model and explain my reasoning before checking my answer.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Torque, rotational inertia and fixed-axis dynamics
Focus: Radius of gyration compresses a mass distribution into one distance
Learning goals:
- Connect angular motion to point speeds and accelerations.
- Calculate signed torques and distinguish force balance from torque balance.
- Derive moment of inertia and use the correct axis.
- Interpret radius of gyration as a mass-weighted rms distance.
- Derive and restrict the perpendicular-axis theorem to planar distributions.
- Solve coupled translation and fixed-axis rotation and check rotational work.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

07 / Torque, rotational inertia and fixed-axis dynamics

The perpendicular-axis theorem comes from a flat geometry

For a thin planar lamina lying in the xy plane, choose perpendicular x and y axes in that plane meeting at an origin, and a z axis through the same point perpendicular to the lamina. A mass element at (x, y, 0) has squared distance x² + y² from z, distance |y| from x and distance |x| from y.

Adding its contributions about x and y therefore gives its contribution about z. Summing or integrating over the flat body gives Iz = Ix + Iy. All three axes must intersect at the same point. They do not have to pass through the centre of mass for this planar identity, but that is often a convenient choice.

The lamina condition matters. For a three-dimensional body, mass elements can have z ≠ 0, and Ix + Iy contains additional 2∫z²dm compared with Iz. The familiar perpendicular-axis sum then fails. Do not apply a planar-disc theorem to a solid sphere merely because both drawings look circular.

I_z=\int(x^2+y^2)\,dm=I_x+I_y\qquad\text{for a lamina in }z=0

For a symmetric planar disc about central diameters, Ix = Iy, so each equals Iz/2. Equality of Ix and Iy requires symmetry; the theorem alone does not provide it.

WORKED EXAMPLE

Use the theorem without inventing symmetry

A uniform rectangular thin lamina has mass 3 kg, width 2 m along x and height 4 m along y. Its central in-plane inertias are Ix = Mb²/12 and Iy = Ma²/12, where a is width and b is height. Find its inertia about the central perpendicular axis.

Choose an approach

The rectangle is planar, so add the two central in-plane inertias. Its unequal side lengths mean those two values differ.

Distance from the x axis depends on the y extent.
$I_x=\frac{3(4^2)}{12}=4\ \mathrm{kg\,m^2}$
Distance from the y axis depends on the x extent.
$I_y=\frac{3(2^2)}{12}=1\ \mathrm{kg\,m^2}$
Add the planar contributions.
$I_z=4+1=5\ \mathrm{kg\,m^2}$

What this gives usThe perpendicular inertia is 5 kg·m². The in-plane values are 4 and 1 kg·m², so assigning each half of Iz would be wrong for this nonsquare rectangle.

Check the result another way

Directly integrating x² + y² over the rectangle separates into the same two independent terms. Both depend on squared side lengths and have units kg·m².

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Worked-example topic: Use the theorem without inventing symmetry
Problem: A uniform rectangular thin lamina has mass 3 kg, width 2 m along x and height 4 m along y. Its central in-plane inertias are Ix = Mb²/12 and Iy = Ma²/12, where a is width and b is height. Find its inertia about the central perpendicular axis.

Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

STOP & THINK

Which body directly permits Iz = Ix + Iy for the stated intersecting perpendicular axes?

A thin lamina entirely in the xy planeA uniform solid sphereAny rigid body if the axes pass through its centre

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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Torque, rotational inertia and fixed-axis dynamics / The perpendicular-axis theorem comes from a flat geometry
Problem: Which body directly permits Iz = Ix + Iy for the stated intersecting perpendicular axes?
Choices:
1. A thin lamina entirely in the xy plane
2. A uniform solid sphere
3. Any rigid body if the axes pass through its centre

Help me choose a model and explain my reasoning before checking my answer.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Torque, rotational inertia and fixed-axis dynamics
Focus: The perpendicular-axis theorem comes from a flat geometry
Learning goals:
- Connect angular motion to point speeds and accelerations.
- Calculate signed torques and distinguish force balance from torque balance.
- Derive moment of inertia and use the correct axis.
- Interpret radius of gyration as a mass-weighted rms distance.
- Derive and restrict the perpendicular-axis theorem to planar distributions.
- Solve coupled translation and fixed-axis rotation and check rotational work.
Equations in this section (LaTeX):
I_z=\int(x^2+y^2)\,dm=I_x+I_y\qquad\text{for a lamina in }z=0
Meaning: For a symmetric planar disc about central diameters, Ix = Iy, so each equals Iz/2. Equality of Ix and Iy requires symmetry; the theorem alone does not provide it.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

08 / Torque, rotational inertia and fixed-axis dynamics

Torque changes angular motion through the axis-specific inertia

For a rigid body rotating about a fixed axis with constant I, the net external torque component about that axis equals Iα. This follows by summing the tangential equations for the body’s mass elements: each contributes mr²α to the required torque.

The equation describes the selected axis component. General three-dimensional rotation can have changing angular-momentum directions and requires a vector treatment beyond the scalar Iα shortcut. Here the axis stays fixed and the body remains rigid, so I is constant.

As with translation, calculate a net quantity. Driving torque, braking torque and frictional torque can oppose each other. A nonzero angular velocity does not require a nonzero net torque; constant angular velocity requires zero net axis torque in this constant-I model. A positive torque can slow a body whose angular velocity is negative.

\tau_{\mathrm{ext,axis}}=I\alpha

The sign of α is the sign of net torque for positive I. Whether speed increases depends also on the sign of ω.

WORKED EXAMPLE

Turn a torque balance into a motion prediction

A wheel has moment of inertia 2 kg·m² about its fixed axle. A driving torque of 8 N·m acts anticlockwise while a 2 N·m resistive torque acts clockwise. Starting from rest, both remain constant for 3 s. Find α, final ω and the angle turned.

Choose an approach

Subtract opposing torques before dividing by I, then use constant-angular-acceleration relations.

The net anticlockwise torque is 6 N·m.
$\alpha=\frac{8-2}{2}=3\ \mathrm{rad/s^2}$
The angular speed rises from zero for 3 s.
$\omega=3(3)=9\ \mathrm{rad/s}$
Integrate angular velocity to obtain displacement.
$\Delta\theta=\frac12(3)(3^2)=13.5\ \mathrm{rad}$

What this gives usThe wheel reaches 9 rad/s after turning 13.5 rad, with constant angular acceleration 3 rad/s².

Check the result another way

The net rotational work is 6 × 13.5 = 81 J. Its final kinetic energy is ½ × 2 × 9² = 81 J, independently checking the result.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Worked-example topic: Turn a torque balance into a motion prediction
Problem: A wheel has moment of inertia 2 kg·m² about its fixed axle. A driving torque of 8 N·m acts anticlockwise while a 2 N·m resistive torque acts clockwise. Starting from rest, both remain constant for 3 s. Find α, final ω and the angle turned.

Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Torque, rotational inertia and fixed-axis dynamics
Focus: Torque changes angular motion through the axis-specific inertia
Learning goals:
- Connect angular motion to point speeds and accelerations.
- Calculate signed torques and distinguish force balance from torque balance.
- Derive moment of inertia and use the correct axis.
- Interpret radius of gyration as a mass-weighted rms distance.
- Derive and restrict the perpendicular-axis theorem to planar distributions.
- Solve coupled translation and fixed-axis rotation and check rotational work.
Equations in this section (LaTeX):
\tau_{\mathrm{ext,axis}}=I\alpha
Meaning: The sign of α is the sign of net torque for positive I. Whether speed increases depends also on the sign of ω.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

09 / Torque, rotational inertia and fixed-axis dynamics

A massive pulley needs a torque as well as a force balance

A hanging mass unwinding a string from a fixed-axis wheel translates while the wheel rotates. If the string does not slip, the mass acceleration and wheel angular acceleration satisfy a = αR. The string tension supplies torque TR to the wheel and also reduces the falling mass’s downward resultant.

The tension is not automatically mg because the mass accelerates. Nor can it be omitted because the string is light: it is the interaction transferring energy and momentum between the falling mass and the rotating wheel. The pulley’s rotational inertia makes the acceleration smaller than free fall.

When a string passes over a massive pulley with two loaded sides, unequal tensions can be necessary to provide its net torque. The equal-tension assumption from the ideal massless pulley no longer follows. Write the torque difference about the axle and combine it with both bodies’ translational equations.

WORKED EXAMPLE

Translation and rotation share one constraint

A 2 kg mass descends while unwinding a light string from a wheel of radius 0.20 m and moment of inertia 0.040 kg·m² about a fixed frictionless axle. The string stays taut and does not slip. Take g = 10 m/s². Find the acceleration and tension.

Choose an approach

Use the mass’s force equation, the wheel’s torque equation and a = αR.

For the falling mass, downward is positive.
$20-T=2a$
For the wheel, the string supplies the torque.
$TR=I\alpha=I\frac aR\quad\Longrightarrow\quad T=\frac I{R^2}a=a$
Substitute the tension relation into the force equation.
$20-a=2a\quad\Longrightarrow\quad a=\frac{20}{3}\ \mathrm{m/s^2},\quad T=\frac{20}{3}\ \mathrm N$

What this gives usThe mass accelerates downward at about 6.67 m/s² and the tension is about 6.67 N. The wheel’s rotational inertia prevents free fall.

Check the result another way

After a descent h from rest, mgh = ½mv² + ½I(v/R)² gives v² = 2mgh/(m + I/R²). Comparing with v² = 2ah returns a = mg/(m + I/R²) = 20/3 m/s².

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Worked-example topic: Translation and rotation share one constraint
Problem: A 2 kg mass descends while unwinding a light string from a wheel of radius 0.20 m and moment of inertia 0.040 kg·m² about a fixed frictionless axle. The string stays taut and does not slip. Take g = 10 m/s². Find the acceleration and tension.

Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Torque, rotational inertia and fixed-axis dynamics
Focus: A massive pulley needs a torque as well as a force balance
Learning goals:
- Connect angular motion to point speeds and accelerations.
- Calculate signed torques and distinguish force balance from torque balance.
- Derive moment of inertia and use the correct axis.
- Interpret radius of gyration as a mass-weighted rms distance.
- Derive and restrict the perpendicular-axis theorem to planar distributions.
- Solve coupled translation and fixed-axis rotation and check rotational work.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

10 / Torque, rotational inertia and fixed-axis dynamics

Torque transfers rotational energy through an angle

A tangential force acting through arc displacement ds = r dθ does work Ftan r dθ = τ dθ. Summing and integrating gives rotational work ∫τ dθ. For a fixed-axis rigid body this equals the change in ½Iω² when τ is the net torque about the axis.

Instantaneous rotational power is τω for the selected axis with consistent signs. A braking torque opposite angular velocity has negative power and removes rotational kinetic energy. An axle support force can be nonzero yet do no work when its point of application is fixed.

These expressions form a useful verification triangle: torque determines α, angular kinematics determines θ and ω, and work–energy compares the endpoints. If the three disagree, check the net torque, radius, inertia axis and whether the acceleration was actually constant. A memorised inertia about the wrong axis can produce dimensionally correct but physically wrong answers.

W_{\mathrm{rot}}=\int\tau\,d\theta=\Delta\left(\frac12I\omega^2\right),\qquad P=\tau\omega

Use radians in work and power relations. Torque has units N·m but describes a different quantity from energy.

STOP & THINK

A constant net torque of 4 N·m turns a fixed-axis rigid body through 3 rad in the torque’s positive direction. How much does its rotational kinetic energy increase?

Your answer (J)Use a decimal, a fraction such as 1/3, or scientific notation.

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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Torque, rotational inertia and fixed-axis dynamics / Torque transfers rotational energy through an angle
Problem: A constant net torque of 4 N·m turns a fixed-axis rigid body through 3 rad in the torque’s positive direction. How much does its rotational kinetic energy increase?
Requested answer unit: J

Help me choose a model and explain my reasoning before checking my answer.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Torque, rotational inertia and fixed-axis dynamics
Focus: Torque transfers rotational energy through an angle
Learning goals:
- Connect angular motion to point speeds and accelerations.
- Calculate signed torques and distinguish force balance from torque balance.
- Derive moment of inertia and use the correct axis.
- Interpret radius of gyration as a mass-weighted rms distance.
- Derive and restrict the perpendicular-axis theorem to planar distributions.
- Solve coupled translation and fixed-axis rotation and check rotational work.
Equations in this section (LaTeX):
W_{\mathrm{rot}}=\int\tau\,d\theta=\Delta\left(\frac12I\omega^2\right),\qquad P=\tau\omega
Meaning: Use radians in work and power relations. Torque has units N·m but describes a different quantity from energy.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

11 / Torque, rotational inertia and fixed-axis dynamics

Choose an axis that removes an unknown, then finish the force balance

A stationary beam has two jobs: avoid translating and avoid rotating. Zero torque about one point alone does not accomplish both. Draw every external force, replace the beam’s weight by its resultant at the centre of gravity, and write the force balance as well as a torque balance. For a uniform beam in uniform gravity the weight acts at its midpoint. A pure applied couple contributes the same torque about every origin.

Choosing the left support as the torque origin makes that support force disappear from the moment equation because its line of action passes through the origin. It has not disappeared from the physical beam. Recover it from the vertical force balance, then check the moments about the other support. This is a useful way to catch a misplaced lever arm or an omitted weight.

An ordinary support underneath a beam can push upward but cannot pull downward. A negative calculated reaction signals loss of contact under that assumed support model. A bolted support can behave differently. State the constraint before accepting a formally solved number.

Original force diagram for the example below. The two supports are 2.4 m apart. Downward forces are 40 N at the midpoint and 24 N at 1.8 m from the left end. Arrows show force directions; lengths are schematic.

WORKED EXAMPLE

Use a second pivot as a check

A uniform 2.4 m beam weighs 40 N. A 24 N load rests 1.8 m from its left end. Simple supports are at both ends. Find both upward reactions.

Choose an approach

Take moments about the left support, then use the total vertical force.

The left reaction has no lever arm about the chosen origin.
$2.4R_B=40(1.2)+24(1.8)$
Divide by the support separation, then balance vertical force.
$R_B=38\ \mathrm N,\qquad R_A=64-38=26\ \mathrm N$

What this gives usBoth positive reactions are compatible with supports underneath the beam.

Check the result another way

About the right end, the moment of the left reaction is 26×2.4=62.4 N·m; the opposite moments are 40×1.2+24×0.6=62.4 N·m.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Worked-example topic: Use a second pivot as a check
Problem: A uniform 2.4 m beam weighs 40 N. A 24 N load rests 1.8 m from its left end. Simple supports are at both ends. Find both upward reactions.

Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

STOP & THINK

A 2 m uniform beam weighs 30 N and is supported at its ends. Add 10 N at 0.5 m from the left end. What is the right reaction in N?

Your answer (N)Use a decimal, a fraction such as 1/3, or scientific notation.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Torque, rotational inertia and fixed-axis dynamics / Choose an axis that removes an unknown, then finish the force balance
Problem: A 2 m uniform beam weighs 30 N and is supported at its ends. Add 10 N at 0.5 m from the left end. What is the right reaction in N?
Requested answer unit: N

Help me choose a model and explain my reasoning before checking my answer.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Torque, rotational inertia and fixed-axis dynamics
Focus: Choose an axis that removes an unknown, then finish the force balance
Learning goals:
- Connect angular motion to point speeds and accelerations.
- Calculate signed torques and distinguish force balance from torque balance.
- Derive moment of inertia and use the correct axis.
- Interpret radius of gyration as a mass-weighted rms distance.
- Derive and restrict the perpendicular-axis theorem to planar distributions.
- Solve coupled translation and fixed-axis rotation and check rotational work.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

12 / Torque, rotational inertia and fixed-axis dynamics

Bring every inertia to the same axis before adding it

The parallel-axis theorem starts from an axis through the body’s centre of mass. It is not a rule that adds Md² to any previously known inertia. If a supplied inertia is already off-centre, first subtract its own centre-of-mass shift, then apply the new one. The axes must be parallel and d is their perpendicular separation.

For a composite rotor, apply this accounting separately to each piece. A small attached mass can be modelled as a point only when its own size is negligible compared with the relevant distances. A mass on the axis contributes no point-mass inertia even though it still contributes to the total mass.

I_A=I_C+Md_A^2,\qquad I_B=I_A-Md_A^2+Md_B^2

C is the centre-of-mass axis parallel to A and B. The difference uses the two squared distances to C, not merely the distance between A and B.

WORKED EXAMPLE

Move an inertia between two parallel axes

A body of mass 4 kg has inertia 1.0 kg·m² about an axis 0.30 m from a parallel centre-of-mass axis. Find its inertia about another parallel axis 0.60 m from the centre of mass.

Choose an approach

Undo the first displacement before adding the second.

Recover the inertia intrinsic to the central axis.
$I_C=1.0-4(0.30)^2=0.64\ \mathrm{kg\,m^2}$
Use the distance from the centre of mass for the new axis.
$I_B=0.64+4(0.60)^2=2.08\ \mathrm{kg\,m^2}$

What this gives usThe requested inertia is 2.08 kg·m².

Check the result another way

Its increase over the original is 4(0.36−0.09)=1.08 kg·m². Adding 4(0.60)² directly to 1.0 would count the first offset twice.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Worked-example topic: Move an inertia between two parallel axes
Problem: A body of mass 4 kg has inertia 1.0 kg·m² about an axis 0.30 m from a parallel centre-of-mass axis. Find its inertia about another parallel axis 0.60 m from the centre of mass.

Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Torque, rotational inertia and fixed-axis dynamics
Focus: Bring every inertia to the same axis before adding it
Learning goals:
- Connect angular motion to point speeds and accelerations.
- Calculate signed torques and distinguish force balance from torque balance.
- Derive moment of inertia and use the correct axis.
- Interpret radius of gyration as a mass-weighted rms distance.
- Derive and restrict the perpendicular-axis theorem to planar distributions.
- Solve coupled translation and fixed-axis rotation and check rotational work.
Equations in this section (LaTeX):
I_A=I_C+Md_A^2,\qquad I_B=I_A-Md_A^2+Md_B^2
Meaning: C is the centre-of-mass axis parallel to A and B. The difference uses the two squared distances to C, not merely the distance between A and B.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

BRING THE IDEAS TOGETHER

What should stay with you

Torque depends on the force line of action and chosen axis.
Moment of inertia weights each mass by squared perpendicular distance.
Radius of gyration satisfies I = Mk² for a stated axis.
Parallel-axis shifts require a centre-of-mass reference axis; perpendicular-axis sums require a lamina.
For fixed-axis constant-I rotation, net axis torque equals Iα.
Rotational work and power use angular displacement and angular velocity in radians.
A convenient pivot removes a force from one torque equation, not from the force balance; all added inertias must refer to the same axis.

RETRIEVE, THEN EXPLAIN

Try it without the examples

Work these out before checking. If an answer surprises you, identify which assumption or step needs another look. A correct choice is most useful when you can explain why the alternatives fail.

SELF CHECK 1

A rigid wheel has I = 0.50 kg·m² about a fixed axis and net torque +3 N·m. What is its signed angular acceleration?

Your answer (rad/s²)Use a decimal, a fraction such as 1/3, or scientific notation.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Torque, rotational inertia and fixed-axis dynamics
Problem: A rigid wheel has I = 0.50 kg·m² about a fixed axis and net torque +3 N·m. What is its signed angular acceleration?
Requested answer unit: rad/s²

Help me choose a model and explain my reasoning before checking my answer.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

SELF CHECK 2

A planar lamina has Ix = 2 kg·m² and Iy = 5 kg·m² about intersecting perpendicular axes in its plane. What is Iz about the perpendicular axis through their intersection?

Your answer (kg·m²)Use a decimal, a fraction such as 1/3, or scientific notation.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Torque, rotational inertia and fixed-axis dynamics
Problem: A planar lamina has Ix = 2 kg·m² and Iy = 5 kg·m² about intersecting perpendicular axes in its plane. What is Iz about the perpendicular axis through their intersection?
Requested answer unit: kg·m²

Help me choose a model and explain my reasoning before checking my answer.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

SELF CHECK 3

Moving a point mass to twice its distance from a fixed axis changes its inertia contribution by what factor?

241/2

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Torque, rotational inertia and fixed-axis dynamics
Problem: Moving a point mass to twice its distance from a fixed axis changes its inertia contribution by what factor?
Choices:
1. 2
2. 4
3. 1/2

Help me choose a model and explain my reasoning before checking my answer.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

SELF CHECK 4

What must be specified before quoting a body’s moment of inertia?

Only its total massThe rotation axis as well as the mass distributionIts current angular speed only

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Torque, rotational inertia and fixed-axis dynamics
Problem: What must be specified before quoting a body’s moment of inertia?
Choices:
1. Only its total mass
2. The rotation axis as well as the mass distribution
3. Its current angular speed only

Help me choose a model and explain my reasoning before checking my answer.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

SELF CHECK 5

Why can two sides of a string over a massive accelerating pulley have different tensions?

A tension difference supplies the pulley’s net torqueNewton’s third law stops applyingThe string must be elastic

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Torque, rotational inertia and fixed-axis dynamics
Problem: Why can two sides of a string over a massive accelerating pulley have different tensions?
Choices:
1. A tension difference supplies the pulley’s net torque
2. Newton’s third law stops applying
3. The string must be elastic

Help me choose a model and explain my reasoning before checking my answer.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

SELF CHECK 6

A 2 kg body has I_C = 0.12 kg·m². Find I about an axis parallel to that central axis and 0.20 m away.

Your answer (kg·m²)Use a decimal, a fraction such as 1/3, or scientific notation.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Torque, rotational inertia and fixed-axis dynamics
Problem: A 2 kg body has I_C = 0.12 kg·m². Find I about an axis parallel to that central axis and 0.20 m away.
Requested answer unit: kg·m²

Help me choose a model and explain my reasoning before checking my answer.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

APPLY THE CHAPTER

Take the ideas into a problem

Try a complete problem next. Commit to a method before revealing a hint, then compare your reasoning with the worked solution.

COME BACK TO THE IDEA

What can you explain later?

Try these from memory at your next study session, then again a few days later. Say your reasoning aloud or sketch it before looking. Adjust the gap to what you can explain.

Next study session

Move a small mass farther from a fixed rotation axis. Why does its contribution to rotational inertia increase?

Compare your reasoning

At the same angular speed it must move faster, requiring more kinetic energy. Its contribution is mr², so doubling its distance quadruples that contribution.

Revisit the explanation →

A few days later: change the situation

Why can a massive pulley require different string tensions on its two sides even without axle friction?

Compare your reasoning

The tension difference supplies the net torque needed to accelerate the pulley’s rotation. Equal tensions would give zero driving torque in the simple two-sided model.

Revisit the explanation →

Compare the physical reason, the assumptions and the conclusion. Your wording can differ. If one part is missing, revisit that section and try explaining it again after a break.

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