Simple harmonic motion

A playground swing keeps returning toward its resting position, but that alone does not make every swing exactly simple harmonic. We need to know how the restoring effect depends on displacement. That detail decides when the elegant sine-wave model earns its place.

This chapter develops a single dynamical idea—acceleration proportional to displacement toward equilibrium—and uses it to connect phase, energy, spring combinations, vertical motion and extended-body pendulums.

What you will learn to do

Recognize the linear restoring law that defines SHM.
Use position and velocity together to determine a phase or initial-state solution.
Derive spring frequencies and identify a shifted equilibrium.
Relate energy fractions to displacement and speed.
State why the simple-pendulum model needs small angles.
Use pivot inertia and centre-of-mass distance for a physical pendulum.
Check how changing a pivot changes both restoring torque and inertia.

Useful before this chapterWork, stored energy and power →Torque, rotational inertia and fixed-axis dynamics →Exponential change, logarithms and controlled approximations →

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: Oscillations: restoring forces, phase and energy
Learning goals:
- Recognize the linear restoring law that defines SHM.
- Use position and velocity together to determine a phase or initial-state solution.
- Derive spring frequencies and identify a shifted equilibrium.
- Relate energy fractions to displacement and speed.
- State why the simple-pendulum model needs small angles.
- Use pivot inertia and centre-of-mass distance for a physical pendulum.
- Check how changing a pivot changes both restoring torque and 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.

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 makes a force restoring? Is every restoring force proportional to displacement?

Compare your reasoning

A restoring force points toward a stable equilibrium. Its magnitude need not be proportional to displacement. Simple harmonic motion requires that additional linear relationship over the range considered.

Revisit the explanation →

01 / Oscillations: restoring forces, phase and energy

Begin with an equilibrium and ask what happens after a displacement

An equilibrium position is one where the resultant force vanishes. It is stable if a small displacement produces a force tending to return the system toward that position. A mass at the lowest point of a smooth bowl is a useful picture: a displacement to either side produces a restoring component of gravity.

Restoring does not mean opposite to velocity. On the outward part of an oscillation, the restoring force slows the object. On the return part, the same force speeds it up. Its direction is determined by displacement from equilibrium, not by whether the object is currently moving left or right.

Simple harmonic motion is the particular oscillation for which acceleration is proportional to displacement and oppositely directed: a = −ω²x. The proportionality coefficient is constant. Many systems approximate this form for small displacements, but a restoring force alone is not enough: a nonlinear restoring law generally produces a different motion.

Choose the equilibrium as x = 0 before testing the law. If a force is written F = F₀ − kq, first find q₀ = F₀/k. Then x = q − q₀ makes the resultant −kx. A constant force can shift the equilibrium without changing the linear stiffness.

\ddot x=-\omega^2x,\qquad \omega>0

Displacement is measured from equilibrium. The units of ω are s⁻¹; it is conventionally expressed in radians per second.

Explain this equation

Read it in words

The acceleration is proportional to displacement from equilibrium and points back toward equilibrium.

Symbols and units

$x$: Signed displacement from the stable equilibrium position.Metres (m).
$\ddot x$: Second time derivative of x: acceleration, not displacement squared.m/s².
$\omega$: Positive angular frequency that sets the oscillation rate.rad/s, with radians dimensionless.

Signs and operations

Two dots mean differentiate with respect to time twice. The minus sign gives restoring acceleration: positive displacement produces negative acceleration. It says nothing by itself about which way the body is moving.

When it applies

This is ideal undamped simple harmonic motion with constant ω and a linear restoring relation. Real pendulums follow it approximately at small angles; large-angle or driven motion needs a different equation.

Make it concrete

If ω = 4 rad/s and x = +0.020 m, acceleration is −0.32 m/s². At equilibrium x = 0, acceleration is zero even though speed is greatest in a nonzero-amplitude oscillation.

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.

Equation (LaTeX): \ddot x=-\omega^2x,\qquad \omega>0
Read it as: The acceleration is proportional to displacement from equilibrium and points back toward equilibrium.
Symbols:
x: Signed displacement from the stable equilibrium position.; Metres (m).
\ddot x: Second time derivative of x: acceleration, not displacement squared.; m/s².
\omega: Positive angular frequency that sets the oscillation rate.; rad/s, with radians dimensionless.
Signs and operations: Two dots mean differentiate with respect to time twice. The minus sign gives restoring acceleration: positive displacement produces negative acceleration. It says nothing by itself about which way the body is moving.
Conditions: This is ideal undamped simple harmonic motion with constant ω and a linear restoring relation. Real pendulums follow it approximately at small angles; large-angle or driven motion needs a different equation.

Help me explain the equation in words, predict how changing a quantity affects the result, and apply it to a small numerical example.

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 mass is to the right of equilibrium and moving left in ideal SHM. Which way is its acceleration?

Right, opposite its velocityLeft, toward equilibriumZero because it is returning

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: Oscillations: restoring forces, phase and energy / Begin with an equilibrium and ask what happens after a displacement
Problem: A mass is to the right of equilibrium and moving left in ideal SHM. Which way is its acceleration?
Choices:
1. Right, opposite its velocity
2. Left, toward equilibrium
3. Zero because it is returning

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: Oscillations: restoring forces, phase and energy
Focus: Begin with an equilibrium and ask what happens after a displacement
Learning goals:
- Recognize the linear restoring law that defines SHM.
- Use position and velocity together to determine a phase or initial-state solution.
- Derive spring frequencies and identify a shifted equilibrium.
- Relate energy fractions to displacement and speed.
- State why the simple-pendulum model needs small angles.
- Use pivot inertia and centre-of-mass distance for a physical pendulum.
- Check how changing a pivot changes both restoring torque and inertia.
Equations in this section (LaTeX):
\ddot x=-\omega^2x,\qquad \omega>0
Meaning: Displacement is measured from equilibrium. The units of ω are s⁻¹; it is conventionally expressed in radians per second.

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 ↑

02 / Oscillations: restoring forces, phase and energy

Position, velocity and acceleration describe different parts of the cycle

A cosine is one solution of the SHM equation. Write x = A cos(ωt + φ), where A is the maximum displacement magnitude and φ specifies the initial state relative to the chosen clock. Differentiating gives v = −Aω sin(ωt + φ) and a = −Aω² cos(ωt + φ).

One full cycle advances phase by 2π radians. Therefore T = 2π/ω and f = 1/T = ω/(2π). Angular frequency is not the number of cycles per second: the factor 2π converts between phase advance and completed cycles.

At x = +A the object is momentarily at rest but has its largest negative acceleration. At x = 0 the acceleration is zero while speed is largest. Zero velocity at a turning point does not imply zero force or equilibrium. The force must change the direction of motion there.

At one intermediate position, the object can be travelling in either direction. Position alone therefore does not determine the phase. The sign of velocity distinguishes the outward and return branches. This is why selecting an inverse cosine without checking velocity can give the wrong time.

x=A\cos(\omega t+\phi),\quad v=-A\omega\sin(\omega t+\phi),\quad a=-\omega^2x

Amplitude A is nonnegative. A different sine or cosine convention describes the same motion if the phase is adjusted consistently.

Read one oscillation at selected times

Inspect a plotted point: t/T = 0

Displacement: 10

Start at maximum positive displacement. The plotted displacement is in centimetres; use the slope sign to infer velocity. Straight segments join sampled values and are not the exact sinusoidal curve. Lines connect the plotted samples.

WORKED EXAMPLE

Set the phase from both position and direction

A particle has amplitude 0.10 m and angular frequency 4.0 rad/s. At t = 0 it passes through equilibrium toward negative x. Write a position function and find the first time it reaches x = −0.10 m.

Choose an approach

Choose a sinusoidal function with zero initial displacement and negative initial velocity.

A negative sine fits both initial conditions.
$x(t)=-0.10\sin(4t)\ \mathrm m$
The first negative extreme occurs when the sine first reaches one.
$4t=\frac{\pi}{2}\quad\Longrightarrow\quad t=\frac{\pi}{8}\ \mathrm s$

What this gives usOne suitable function is x = −0.10 sin(4t) m. It reaches the negative extreme after π/8 s, approximately 0.393 s.

Check the result another way

Differentiation gives v(0) = −0.40 m/s. The period is π/2 s, so the first extreme is one quarter of a period after the stated equilibrium crossing.

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: Set the phase from both position and direction
Problem: A particle has amplitude 0.10 m and angular frequency 4.0 rad/s. At t = 0 it passes through equilibrium toward negative x. Write a position function and find the first time it reaches x = −0.10 m.

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: Oscillations: restoring forces, phase and energy
Focus: Position, velocity and acceleration describe different parts of the cycle
Learning goals:
- Recognize the linear restoring law that defines SHM.
- Use position and velocity together to determine a phase or initial-state solution.
- Derive spring frequencies and identify a shifted equilibrium.
- Relate energy fractions to displacement and speed.
- State why the simple-pendulum model needs small angles.
- Use pivot inertia and centre-of-mass distance for a physical pendulum.
- Check how changing a pivot changes both restoring torque and inertia.
Equations in this section (LaTeX):
x=A\cos(\omega t+\phi),\quad v=-A\omega\sin(\omega t+\phi),\quad a=-\omega^2x
Meaning: Amplitude A is nonnegative. A different sine or cosine convention describes the same motion if the phase is adjusted consistently.

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 / Oscillations: restoring forces, phase and energy

Recover amplitude without guessing a phase

Eliminate the phase between the position and velocity equations using sin²θ + cos²θ = 1. The result v² = ω²(A² − x²) determines speed at a displacement. It does not determine the direction of velocity; retain a separate sign from the motion.

For known initial position x₀ and velocity v₀, amplitude is √[x₀² + (v₀/ω)²]. The ratio v₀/ω has units of length, so adding its square to x₀² is dimensionally consistent. Initial speed contributes to amplitude even if the release point is not an extreme.

A particularly convenient solution is x(t) = x₀ cos ωt + (v₀/ω) sin ωt. Substituting t = 0 and differentiating verifies both initial conditions immediately. This form avoids inverse-trigonometric branch mistakes when no explicit phase angle is needed.

A=\sqrt{x_0^2+\frac{v_0^2}{\omega^2}},\qquad x(t)=x_0\cos\omega t+\frac{v_0}{\omega}\sin\omega t

These expressions assume a constant ω and displacement measured from the correct equilibrium.

WORKED EXAMPLE

Combine an initial displacement with an initial push

An ideal oscillator has ω = 5 rad/s, x₀ = 0.060 m and v₀ = −0.40 m/s. Find its amplitude and maximum speed.

Choose an approach

Use the phase-independent amplitude relation, then multiply by ω.

Convert the initial velocity into the corresponding length scale.
$v_0/\omega=-0.40/5=-0.080\ \mathrm m$
Combine the squared position and velocity contributions.
$A=\sqrt{0.060^2+0.080^2}=0.100\ \mathrm m$
Maximum speed occurs at equilibrium.
$v_{\max}=\omega A=0.50\ \mathrm{m/s}$

What this gives usThe amplitude is 0.100 m and the maximum speed is 0.50 m/s. The negative initial velocity means the mass initially approaches equilibrium from positive x.

Check the result another way

At x = 0.060 m, ω√(A² − x²) = 5 × 0.080 = 0.40 m/s, matching the given speed.

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: Combine an initial displacement with an initial push
Problem: An ideal oscillator has ω = 5 rad/s, x₀ = 0.060 m and v₀ = −0.40 m/s. Find its amplitude and maximum speed.

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: Oscillations: restoring forces, phase and energy
Focus: Recover amplitude without guessing a phase
Learning goals:
- Recognize the linear restoring law that defines SHM.
- Use position and velocity together to determine a phase or initial-state solution.
- Derive spring frequencies and identify a shifted equilibrium.
- Relate energy fractions to displacement and speed.
- State why the simple-pendulum model needs small angles.
- Use pivot inertia and centre-of-mass distance for a physical pendulum.
- Check how changing a pivot changes both restoring torque and inertia.
Equations in this section (LaTeX):
A=\sqrt{x_0^2+\frac{v_0^2}{\omega^2}},\qquad x(t)=x_0\cos\omega t+\frac{v_0}{\omega}\sin\omega t
Meaning: These expressions assume a constant ω and displacement measured from the correct equilibrium.

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 ↑

04 / Oscillations: restoring forces, phase and energy

Derive the spring period from force and inertia

For a mass m attached to a massless ideal spring of stiffness k on a frictionless horizontal surface, Hooke’s law gives F = −kx. Newton’s law then gives ẍ = −(k/m)x. Comparing with the SHM equation identifies ω² = k/m and T = 2π√(m/k).

A larger mass is harder to accelerate and gives a longer period. A stiffer spring produces a larger restoring force for the same displacement and gives a shorter period. These explanations are more useful than remembering which quantity is on top of a square root.

The period is independent of amplitude only within the linear, undamped model. A real spring may cease to obey Hooke’s law at large extension, and its own mass can contribute inertia. Friction or fluid drag can dissipate energy. State which of these effects has been neglected before applying the ideal period.

When springs act on the same displacement in parallel, their restoring forces add, so the effective stiffness is k₁ + k₂. For two massless springs in series carrying the same force, their extensions add, so 1/k_eff = 1/k₁ + 1/k₂. These rules follow from compatibility and force balance rather than from a diagram’s visual arrangement alone.

\omega=\sqrt{\frac{k_{\mathrm{eff}}}{m}},\qquad T=2\pi\sqrt{\frac{m}{k_{\mathrm{eff}}}}

Find the restoring force for the actual mass displacement before choosing k_eff.

WORKED EXAMPLE

Determine the stiffness seen by the moving mass

A 0.50 kg mass is connected between two fixed walls by horizontal ideal springs of stiffnesses 30 N/m and 20 N/m. At equilibrium both springs remain in their linear range. The mass is displaced slightly to the right. Find the angular frequency.

Choose an approach

Each spring’s change of force opposes the same mass displacement, so their restoring contributions add.

A rightward displacement stretches one spring and compresses the other, with both changes restoring.
$F_{\mathrm{net}}=-(30+20)x=-50x$
Identify the squared angular frequency from Newton’s law.
$\omega^2=50/0.50=100\ \mathrm{s^{-2}}$

What this gives usThe angular frequency is 10 rad/s and the period is 2π/10 s. This arrangement has effective stiffness 50 N/m.

Check the result another way

For a 1 cm displacement, the net restoring force is 0.50 N and acceleration magnitude is 1.0 m/s², equal to ω²|x|.

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: Determine the stiffness seen by the moving mass
Problem: A 0.50 kg mass is connected between two fixed walls by horizontal ideal springs of stiffnesses 30 N/m and 20 N/m. At equilibrium both springs remain in their linear range. The mass is displaced slightly to the right. Find the angular frequency.

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: Oscillations: restoring forces, phase and energy
Focus: Derive the spring period from force and inertia
Learning goals:
- Recognize the linear restoring law that defines SHM.
- Use position and velocity together to determine a phase or initial-state solution.
- Derive spring frequencies and identify a shifted equilibrium.
- Relate energy fractions to displacement and speed.
- State why the simple-pendulum model needs small angles.
- Use pivot inertia and centre-of-mass distance for a physical pendulum.
- Check how changing a pivot changes both restoring torque and inertia.
Equations in this section (LaTeX):
\omega=\sqrt{\frac{k_{\mathrm{eff}}}{m}},\qquad T=2\pi\sqrt{\frac{m}{k_{\mathrm{eff}}}}
Meaning: Find the restoring force for the actual mass displacement before choosing k_eff.

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 ↑

05 / Oscillations: restoring forces, phase and energy

Gravity shifts a vertical spring’s equilibrium

Let q be the downward extension of a vertical spring from its natural length. The mass experiences mg downward and kq upward, so m q̈ = mg − kq. Static equilibrium occurs at q₀ = mg/k.

Define x = q − q₀. Since q₀ is constant, ẍ = q̈, and the equation becomes m ẍ = −kx. The small-oscillation frequency is therefore the same √(k/m) as for the ideal horizontal spring. Gravity determines the equilibrium extension, not an additional linear stiffness.

Spring energy alone is not the total position-dependent energy in the vertical problem. The combination ½kq² − mgq can be written ½k(q − q₀)² plus a constant. This explains why the effective oscillator potential is ½kx² when x is measured from equilibrium.

For a spring or cord that cannot push or remain taut, check whether the proposed amplitude carries the system into a slack region. The ideal linear equation assumes its force law stays valid throughout the motion.

WORKED EXAMPLE

Release from natural length without using the wrong origin

A 0.20 kg mass is attached to an ideal vertical spring with k = 50 N/m. It is released from rest when the spring has its natural length. Take g = 10 m/s² and assume the spring remains described by the ideal force law. Find the equilibrium extension, amplitude about equilibrium and greatest extension.

Choose an approach

Measure the oscillation from the shifted equilibrium, not from the natural length.

The static extension follows from the weight balance.
$q_0=mg/k=0.20(10)/50=0.040\ \mathrm m$
At release, x₀ = q − q₀ = −q₀ and velocity is zero.
$A=|x_0|=0.040\ \mathrm m$
The lower turning point lies one amplitude below equilibrium.
$q_{\max}=q_0+A=0.080\ \mathrm m$

What this gives usThe equilibrium extension and amplitude are each 4.0 cm, while the greatest extension is 8.0 cm. Natural length is a turning point in this release, not an equilibrium.

Check the result another way

From release to the lower turning point, mgq_max = ½kq_max² = 0.16 J, so the gravitational loss matches the spring-energy increase.

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: Release from natural length without using the wrong origin
Problem: A 0.20 kg mass is attached to an ideal vertical spring with k = 50 N/m. It is released from rest when the spring has its natural length. Take g = 10 m/s² and assume the spring remains described by the ideal force law. Find the equilibrium extension, amplitude about equilibrium and greatest extension.

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

For an ideal vertical mass–spring oscillator, changing g while keeping m and k fixed changes which quantity?

The equilibrium extension, but not √(k/m)The angular frequency, but not equilibriumNeither quantity

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: Oscillations: restoring forces, phase and energy / Gravity shifts a vertical spring’s equilibrium
Problem: For an ideal vertical mass–spring oscillator, changing g while keeping m and k fixed changes which quantity?
Choices:
1. The equilibrium extension, but not √(k/m)
2. The angular frequency, but not equilibrium
3. Neither quantity

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: Oscillations: restoring forces, phase and energy
Focus: Gravity shifts a vertical spring’s equilibrium
Learning goals:
- Recognize the linear restoring law that defines SHM.
- Use position and velocity together to determine a phase or initial-state solution.
- Derive spring frequencies and identify a shifted equilibrium.
- Relate energy fractions to displacement and speed.
- State why the simple-pendulum model needs small angles.
- Use pivot inertia and centre-of-mass distance for a physical pendulum.
- Check how changing a pivot changes both restoring torque and inertia.
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.

Back to chapter contents ↑

06 / Oscillations: restoring forces, phase and energy

Energy links all positions without solving for time

For an ideal spring oscillator measured from equilibrium, total oscillation energy is ½kA². At displacement x, the potential part is ½kx² and the kinetic part is ½k(A² − x²). Their sum remains constant while energy passes between motion and deformation.

At a turning point, kinetic energy is zero but force and acceleration are not. At equilibrium, the chosen oscillator potential is zero and kinetic energy is greatest. The object does not stop there because its existing velocity carries it past equilibrium before the restoring force reverses direction.

Kinetic and potential energies each repeat twice per displacement cycle: the two opposite extremes have equal potential energies and the two equilibrium crossings have equal kinetic energies. An energy graph can therefore have period T/2 even though the position has period T.

Use the lab to separate position, speed and force. Pause at an extreme, at equilibrium and at half the amplitude. Predict the signs first, then inspect the readouts. Changing amplitude should change maximum speed and total energy while leaving the ideal period fixed if m and k stay fixed.

TRY THE IDEA

Follow the motion and energy together

Move through the cycle manually. At x = A/2, predict U/E and K/E before checking the bars. Compare the two times that share the same displacement: the speed magnitude matches, but its direction differs.

MOVE THROUGH A CYCLESpring + cart

Where does the energy go?

1 kg cart · 100 N/m spring · 10 cm amplitude · no friction

Position in the cycle 0%

Spring energy0.500 J

Kinetic energy0.000 J

Displacement from equilibrium10.0 cm

Velocity (right positive)0.00 m/s

Spring force (right positive)-10.00 N

At a turning point the cart pauses. All 0.500 J is stored in the spring.

Phase playback is slowed for inspection. Physical period: 0.628 s.

WORKED EXAMPLE

Translate an energy fraction into a position

At what displacement magnitudes does an ideal spring oscillator have kinetic energy three times its potential energy?

Choose an approach

Express both energies as fractions of the fixed total.

If K = 3U, then E = K + U = 4U.
$\frac{U}{E}=\frac14$
The potential-energy fraction equals the squared displacement fraction.
$\frac{x^2}{A^2}=\frac14\quad\Longrightarrow\quad |x|=\frac A2$

What this gives usThe positions are x = +A/2 and x = −A/2. Each is visited twice per cycle with opposite velocities.

Check the result another way

At either position, v² = ω²(A² − A²/4) = 3ω²A²/4, giving K/E = 3/4 and U/E = 1/4.

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: Translate an energy fraction into a position
Problem: At what displacement magnitudes does an ideal spring oscillator have kinetic energy three times its potential energy?

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: Oscillations: restoring forces, phase and energy
Focus: Energy links all positions without solving for time
Learning goals:
- Recognize the linear restoring law that defines SHM.
- Use position and velocity together to determine a phase or initial-state solution.
- Derive spring frequencies and identify a shifted equilibrium.
- Relate energy fractions to displacement and speed.
- State why the simple-pendulum model needs small angles.
- Use pivot inertia and centre-of-mass distance for a physical pendulum.
- Check how changing a pivot changes both restoring torque and inertia.
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.

Back to chapter contents ↑

07 / Oscillations: restoring forces, phase and energy

A pendulum becomes harmonic through a small-angle approximation

For a simple pendulum, model the bob as a point mass at distance L from a fixed frictionless pivot, supported by a massless rigid link or a taut string. The tangential restoring force is −mg sin θ. Since tangential acceleration is Lθ̈, the exact ideal angular equation is θ̈ = −(g/L) sin θ.

For sufficiently small angles measured in radians, sin θ ≈ θ. The equation then has SHM form, with angular frequency √(g/L) and period 2π√(L/g). The approximation is in the force law, not in the definition of period or in Newton’s law.

At larger angular amplitudes, the exact motion remains periodic in the ideal model but is not sinusoidal, and its period depends on amplitude. A question saying pendulum does not automatically justify the small-angle formula.

The bob’s mass cancels because both its gravitational restoring torque and inertia are proportional to mass. The length is the pivot-to-bob distance in the point-mass model. Changing local effective gravity, moving the support or allowing the string to go slack requires revisiting the model.

\ddot\theta=-\frac gL\sin\theta\approx-\frac gL\theta,\qquad T\approx2\pi\sqrt{\frac Lg}

The period expression applies to small angular oscillations about the stable downward equilibrium.

STOP & THINK

Why is the ideal small-angle pendulum period independent of bob mass?

Gravity does not act on the bobBoth restoring torque and inertia scale with massThe tension always equals the weight

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: Oscillations: restoring forces, phase and energy / A pendulum becomes harmonic through a small-angle approximation
Problem: Why is the ideal small-angle pendulum period independent of bob mass?
Choices:
1. Gravity does not act on the bob
2. Both restoring torque and inertia scale with mass
3. The tension always equals the weight

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: Oscillations: restoring forces, phase and energy
Focus: A pendulum becomes harmonic through a small-angle approximation
Learning goals:
- Recognize the linear restoring law that defines SHM.
- Use position and velocity together to determine a phase or initial-state solution.
- Derive spring frequencies and identify a shifted equilibrium.
- Relate energy fractions to displacement and speed.
- State why the simple-pendulum model needs small angles.
- Use pivot inertia and centre-of-mass distance for a physical pendulum.
- Check how changing a pivot changes both restoring torque and inertia.
Equations in this section (LaTeX):
\ddot\theta=-\frac gL\sin\theta\approx-\frac gL\theta,\qquad T\approx2\pi\sqrt{\frac Lg}
Meaning: The period expression applies to small angular oscillations about the stable downward equilibrium.

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 / Oscillations: restoring forces, phase and energy

An extended body needs its moment of inertia about the pivot

A swinging rigid body has mass distributed over many distances from the pivot. Its gravitational torque is −mgd sin θ, where d is the pivot-to-centre-of-mass distance. The rotational equation is I_p θ̈ = −mgd sin θ, with I_p taken about the actual fixed pivot axis.

At small angles, the period is 2π√(I_p/(mgd)). Replacing the body by a point mass at its centre of mass would incorrectly use md² for its entire inertia. An extended body also has its own centre-of-mass moment of inertia, so the parallel-axis theorem gives I_p = I_cm + md².

An equivalent simple-pendulum length is ℓ_eq = I_p/(md). This is a dynamical length and need not equal the body’s full length or the centre-of-mass distance. The units are metres, and substituting it into 2π√(ℓ_eq/g) reproduces the physical-pendulum period.

If the pivot passes through the centre of mass, d = 0 and gravity supplies no restoring torque about that axis. The small-oscillation formula does not then describe a finite-frequency gravitational oscillator. As a pivot approaches the centre of mass for a body with nonzero I_cm, the period grows rather than tending to zero.

T\approx2\pi\sqrt{\frac{I_p}{mgd}},\qquad I_p=I_{\mathrm{cm}}+md^2

The pivot is fixed, the axis is horizontal, and d is positive for the stable centre-of-mass-below-pivot equilibrium.

WORKED EXAMPLE

Use the full inertia of a swinging rod

A uniform thin rod of length 0.90 m swings through small angles about a fixed horizontal axis through one end, perpendicular to the rod. Take g = 10 m/s². Find its equivalent pendulum length and angular frequency.

Choose an approach

Use the rod’s end-axis inertia and the location of its centre of mass.

The centre of mass is halfway along the rod.
$d=L/2,\qquad I_p=\frac13mL^2$
Form the equivalent length from inertia divided by md.
$\ell_{\mathrm{eq}}=\frac{mL^2/3}{mL/2}=\frac{2L}{3}=0.60\ \mathrm m$
Use the corresponding gravity-to-length ratio.
$\omega=\sqrt{g/\ell_{\mathrm{eq}}}=\sqrt{50/3}\ \mathrm{rad/s}$

What this gives usThe equivalent length is 0.60 m and ω ≈ 4.08 rad/s. The period is about 1.54 s. The rod behaves neither like a bob at 0.90 m nor like a bob at its 0.45 m centre-of-mass distance.

Check the result another way

The parallel-axis theorem gives I_p = mL²/12 + m(L/2)² = mL²/3, independently confirming the inertia used.

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 full inertia of a swinging rod
Problem: A uniform thin rod of length 0.90 m swings through small angles about a fixed horizontal axis through one end, perpendicular to the rod. Take g = 10 m/s². Find its equivalent pendulum length and angular frequency.

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: Oscillations: restoring forces, phase and energy
Focus: An extended body needs its moment of inertia about the pivot
Learning goals:
- Recognize the linear restoring law that defines SHM.
- Use position and velocity together to determine a phase or initial-state solution.
- Derive spring frequencies and identify a shifted equilibrium.
- Relate energy fractions to displacement and speed.
- State why the simple-pendulum model needs small angles.
- Use pivot inertia and centre-of-mass distance for a physical pendulum.
- Check how changing a pivot changes both restoring torque and inertia.
Equations in this section (LaTeX):
T\approx2\pi\sqrt{\frac{I_p}{mgd}},\qquad I_p=I_{\mathrm{cm}}+md^2
Meaning: The pivot is fixed, the axis is horizontal, and d is positive for the stable centre-of-mass-below-pivot equilibrium.

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 / Oscillations: restoring forces, phase and energy

Moving the pivot changes both inertia and restoring torque

Write I_cm = mk_g², where k_g is the radius of gyration about the parallel centre-of-mass axis. Then the equivalent length of a physical pendulum is ℓ_eq = d + k_g²/d. Increasing d strengthens the gravitational restoring torque, but it also changes the pivot inertia. Neither effect should be considered alone.

For a given rigid body with allowed pivot locations, the expression has its minimum at d = k_g. This follows by differentiation or from d + k_g²/d ≥ 2k_g for d > 0. Whether that pivot location is physically available depends on the body and support geometry.

A timing experiment can infer inertia only if the small-angle and fixed-pivot assumptions are adequate. Measure several periods to reduce reaction-time error, keep release angles small, avoid giving a push, and distinguish the measured pivot-to-centre-of-mass distance from the body’s total length. Repeated timings reduce scatter but cannot correct a misplaced pivot measurement.

WORKED EXAMPLE

Compare two pivots without calculating a period twice

A rigid body has radius of gyration k_g about its horizontal centre-of-mass axis. Compare small-oscillation periods for pivot distances d₁ = k_g/2 and d₂ = 2k_g along a suitable line above its centre of mass.

Choose an approach

Compare equivalent lengths d + k_g²/d.

Evaluate the equivalent length at the nearer pivot.
$\ell_1=\frac{k_g}{2}+2k_g=\frac52k_g$
Evaluate the equivalent length at the farther pivot.
$\ell_2=2k_g+\frac{k_g}{2}=\frac52k_g$

What this gives usThe two periods are equal even though their pivot distances differ by a factor of four. Each equivalent length is 2.5k_g.

Check the result another way

Both exceed the minimum equivalent length 2k_g at d = k_g, so both periods exceed the minimum available in the unrestricted model.

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: Compare two pivots without calculating a period twice
Problem: A rigid body has radius of gyration k_g about its horizontal centre-of-mass axis. Compare small-oscillation periods for pivot distances d₁ = k_g/2 and d₂ = 2k_g along a suitable line above its 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.

STOP & THINK

A body has k_g = 0.12 m and its pivot is d = 0.24 m above its centre of mass. What is its equivalent pendulum length?

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: Oscillations: restoring forces, phase and energy / Moving the pivot changes both inertia and restoring torque
Problem: A body has k_g = 0.12 m and its pivot is d = 0.24 m above its centre of mass. What is its equivalent pendulum length?
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: Oscillations: restoring forces, phase and energy
Focus: Moving the pivot changes both inertia and restoring torque
Learning goals:
- Recognize the linear restoring law that defines SHM.
- Use position and velocity together to determine a phase or initial-state solution.
- Derive spring frequencies and identify a shifted equilibrium.
- Relate energy fractions to displacement and speed.
- State why the simple-pendulum model needs small angles.
- Use pivot inertia and centre-of-mass distance for a physical pendulum.
- Check how changing a pivot changes both restoring torque and inertia.
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.

Back to chapter contents ↑

10 / Oscillations: restoring forces, phase and energy

Decide what the ideal model leaves out

Damping removes mechanical energy and usually reduces the amplitude with time. A driven oscillator also receives energy from an external time-dependent force. Its steady response depends on driving frequency and damping; a large response near resonance is not described by an undriven constant-amplitude solution.

The useful route is always the same: identify the equilibrium, choose a displacement coordinate, derive the resultant force or torque and inspect its dependence on displacement. A linear restoring term gives the ideal natural frequency when divided by the appropriate mass or inertia.

For a more complicated stable equilibrium, a smooth potential can often be approximated locally by a quadratic curve. The positive curvature acts as an effective stiffness. This explains why SHM appears in many small oscillations, while also explaining why the approximation becomes less accurate when excursions explore strongly nonlinear parts of the potential.

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: Oscillations: restoring forces, phase and energy
Focus: Decide what the ideal model leaves out
Learning goals:
- Recognize the linear restoring law that defines SHM.
- Use position and velocity together to determine a phase or initial-state solution.
- Derive spring frequencies and identify a shifted equilibrium.
- Relate energy fractions to displacement and speed.
- State why the simple-pendulum model needs small angles.
- Use pivot inertia and centre-of-mass distance for a physical pendulum.
- Check how changing a pivot changes both restoring torque and inertia.
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 ↑

BRING THE IDEAS TOGETHER

What should stay with you

SHM requires a = −ω²x with x measured from equilibrium.
A turning point has zero speed and maximum restoring-acceleration magnitude.
Initial position and velocity together determine amplitude and phase.
An ideal spring has ω² = k_eff/m; a constant force can shift equilibrium without changing stiffness.
Energy fractions depend on x²/A², while velocity direction requires additional information.
A physical pendulum uses I about the pivot, not just md².
Small-angle, linearity and negligible-damping assumptions must be checked.

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

An ideal oscillator has mass 0.80 kg and effective spring stiffness 20 N/m. What is its angular frequency?

Your answer (rad/s)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: Oscillations: restoring forces, phase and energy
Problem: An ideal oscillator has mass 0.80 kg and effective spring stiffness 20 N/m. What is its angular frequency?
Requested answer unit: rad/s

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

At x = 0.60A in a spring oscillator, what fraction of the total energy is kinetic?

Your answerUse 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: Oscillations: restoring forces, phase and energy
Problem: At x = 0.60A in a spring oscillator, what fraction of the total energy is kinetic?
Requested answer unit: 

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

At a positive turning point of undamped SHM, which statement is correct?

Velocity and acceleration are both zeroVelocity is zero and acceleration is negativeAcceleration is zero and speed is maximum

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: Oscillations: restoring forces, phase and energy
Problem: At a positive turning point of undamped SHM, which statement is correct?
Choices:
1. Velocity and acceleration are both zero
2. Velocity is zero and acceleration is negative
3. Acceleration is zero and speed is maximum

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

Why can an extended body not generally be replaced by all its mass at its centre of mass when computing a pendulum period?

Its pivot inertia also includes I_cmGravity acts only at the pivotIts centre of mass has no motion

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: Oscillations: restoring forces, phase and energy
Problem: Why can an extended body not generally be replaced by all its mass at its centre of mass when computing a pendulum period?
Choices:
1. Its pivot inertia also includes I_cm
2. Gravity acts only at the pivot
3. Its centre of mass has no motion

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

Which change multiplies the ideal spring period by two?

Doubling amplitudeMultiplying mass by four at fixed stiffnessMultiplying stiffness by four at fixed mass

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: Oscillations: restoring forces, phase and energy
Problem: Which change multiplies the ideal spring period by two?
Choices:
1. Doubling amplitude
2. Multiplying mass by four at fixed stiffness
3. Multiplying stiffness by four at fixed mass

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.

A swinging rod is not a point bobIntermediate · Free sample · Hints and guided reasoning How quickly does the spring cycle?Intermediate · Free sample · Hints and guided reasoning How fast is an oscillator halfway to an end?Intermediate · Hints and guided reasoning What changes when gravity becomes weaker?Beginner · Hints and guided reasoning Speed through equilibriumIntermediate · Hints and guided reasoning Share the energy at half amplitudeIntermediate · Hints and guided reasoning First arrival at equilibriumIntermediate · Hints and guided reasoning A string point moves differently from the waveIntermediate · Hints and guided reasoning

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

At the turning point of an ideal spring oscillator, explain the speed, acceleration and energy store.

Compare your reasoning

Speed is zero, acceleration points toward equilibrium and has its largest magnitude, and the mechanical energy is in the spring store. Zero speed does not imply equilibrium.

Revisit the explanation →

A few days later: change the situation

If an ideal linear spring oscillator starts with twice the amplitude, what changes about its period and total energy?

Compare your reasoning

Its period is unchanged for the same mass and spring constant, while total energy quadruples. The claim about period depends on the force staying linear over the larger motion.

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.

Back to chapter contents ↑