C.1 Simple harmonic motion

Explain why the magnitude of the emf is related to the amplitude of the ground movement.

Explain why the magnitude of the emf is related to the amplitude of the ground movement.

Explain why the magnitude of the emf is related to the amplitude of the ground movement.

The charges Q are replaced by neutral masses M and the charge q by a neutral mass m. The mass m is displaced away from C by a small distance $x$ and released. Discuss whether the motion of m will be the same as that of q.

The charges Q are replaced by neutral masses M and the charge q by a neutral mass m. The mass m is displaced away from C by a small distance $x$ and released. Discuss whether the motion of m will be the same as that of q.

The charges Q are replaced by neutral masses M and the charge q by a neutral mass m. The mass m is displaced away from C by a small distance $x$ and released. Discuss whether the motion of m will be the same as that of q.

The magnitude of the net force on q is given by $\frac{32kQq}{D^3}x$. Explain why the charge q will execute simple harmonic oscillations about C.

The magnitude of the net force on q is given by $\frac{32kQq}{D^3}x$. Explain why the charge q will execute simple harmonic oscillations about C.

The magnitude of the net force on q is given by $\frac{32kQq}{D^3}x$. Explain why the charge q will execute simple harmonic oscillations about C.

Show that the phase difference between the oscillations of the two corks is $\pi$ radians.

Show that the phase difference between the oscillations of the two corks is $\pi$ radians.

Show that the phase difference between the oscillations of the two corks is $\pi$ radians.

The mass of the charge q is 0.025 kg.

Calculate the angular frequency of the oscillations using the data in (a)(ii) and the expression in (b)(i).

The mass of the charge q is 0.025 kg.

Calculate the angular frequency of the oscillations using the data in (a)(ii) and the expression in (b)(i).

The mass of the charge q is 0.025 kg.

Calculate the angular frequency of the oscillations using the data in (a)(ii) and the expression in (b)(i).

A particle undergoes simple harmonic motion of period $T$. At time $t=0$ the particle is at its equilibrium position.

What is $t$ when the particle is at its greatest distance from the equilibrium position?

A.  $\frac{T}{8}$

B.  $\frac{T}{2}$

C.  $\frac{3T}{4}$

D.  $T$

A particle undergoes simple harmonic motion of period $T$. At time $t=0$ the particle is at its equilibrium position.

What is $t$ when the particle is at its greatest distance from the equilibrium position?

A.  $\frac{T}{8}$

B.  $\frac{T}{2}$

C.  $\frac{3T}{4}$

D.  $T$

A particle undergoes simple harmonic motion of period $T$. At time $t=0$ the particle is at its equilibrium position.

What is $t$ when the particle is at its greatest distance from the equilibrium position?

A.  $\frac{T}{8}$

B.  $\frac{T}{2}$

C.  $\frac{3T}{4}$

D.  $T$

A particle undergoes simple harmonic motion of period $T$. At time $t=0$ the particle is at its equilibrium position.

What is $t$ when the particle is at its greatest distance from the equilibrium position?

A.  $\frac{T}{8}$

B.  $\frac{T}{2}$

C.  $\frac{3T}{4}$

D.  $T$

A particle undergoes simple harmonic motion of period $T$. At time $t=0$ the particle is at its equilibrium position.

What is $t$ when the particle is at its greatest distance from the equilibrium position?

A.  $\frac{T}{8}$

B.  $\frac{T}{2}$

C.  $\frac{3T}{4}$

D.  $T$

A particle undergoes simple harmonic motion of period $T$. At time $t=0$ the particle is at its equilibrium position.

What is $t$ when the particle is at its greatest distance from the equilibrium position?

A.  $\frac{T}{8}$

B.  $\frac{T}{2}$

C.  $\frac{3T}{4}$

D.  $T$

A particle undergoes simple harmonic motion of period $T$. At time $t=0$ the particle is at its equilibrium position.

What is $t$ when the particle is at its greatest distance from the equilibrium position?

A.  $\frac{T}{8}$

B.  $\frac{T}{2}$

C.  $\frac{3T}{4}$

D.  $T$

A particle undergoes simple harmonic motion of period $T$. At time $t=0$ the particle is at its equilibrium position.

What is $t$ when the particle is at its greatest distance from the equilibrium position?

A.  $\frac{T}{8}$

B.  $\frac{T}{2}$

C.  $\frac{3T}{4}$

D.  $T$

A particle undergoes simple harmonic motion of period $T$. At time $t=0$ the particle is at its equilibrium position.

What is $t$ when the particle is at its greatest distance from the equilibrium position?

A.  $\frac{T}{8}$

B.  $\frac{T}{2}$

C.  $\frac{3T}{4}$

D.  $T$

A particle undergoes simple harmonic motion of period $T$. At time $t=0$ the particle is at its equilibrium position.

What is $t$ when the particle is at its greatest distance from the equilibrium position?

A.  $\frac{T}{8}$

B.  $\frac{T}{2}$

C.  $\frac{3T}{4}$

D.  $T$

The four pendulums shown have been cut from the same uniform sheet of board. They are attached to the ceiling with strings of equal length.

Which pendulum has the shortest period?

The four pendulums shown have been cut from the same uniform sheet of board. They are attached to the ceiling with strings of equal length.

Which pendulum has the shortest period?

An object at the end of a spring oscillates vertically with simple harmonic motion (shm). The graph shows the variation with time $t$ of the displacement $x$ of the object.

What is the velocity of the object?

A. $-\frac{2\pi A}{T}\sin \left(\frac{\pi t}{T}\right)$

B. $\frac{2\pi A}{T}\sin \left(\frac{\pi t}{T}\right)$

C. $-\frac{2\pi A}{T}\cos \left(\frac{\pi t}{T}\right)$

D. $\frac{2\pi A}{T}\cos \left(\frac{\pi t}{T}\right)$ 

An object at the end of a spring oscillates vertically with simple harmonic motion (shm). The graph shows the variation with time $t$ of the displacement $x$ of the object.

What is the velocity of the object?

A. $-\frac{2\pi A}{T}\sin \left(\frac{\pi t}{T}\right)$

B. $\frac{2\pi A}{T}\sin \left(\frac{\pi t}{T}\right)$

C. $-\frac{2\pi A}{T}\cos \left(\frac{\pi t}{T}\right)$

D. $\frac{2\pi A}{T}\cos \left(\frac{\pi t}{T}\right)$ 

A particle performs simple harmonic motion (shm). What is the phase difference between the displacement and the acceleration of the particle?

A. 0

B. $\frac{\pi }{2}$

C. $\pi$

D. $\frac{3\pi }{2}$

A particle performs simple harmonic motion (shm). What is the phase difference between the displacement and the acceleration of the particle?

A. 0

B. $\frac{\pi }{2}$

C. $\pi$

D. $\frac{3\pi }{2}$

Which graph shows the variation with time t of the kinetic energy (KE) of an object undergoing simple harmonic motion (shm) of period T?

Which graph shows the variation with time t of the kinetic energy (KE) of an object undergoing simple harmonic motion (shm) of period T?

Object P moves vertically with simple harmonic motion (shm). Object Q moves in a vertical circle with a uniform speed. P and Q have the same time period T. When P is at the top of its motion, Q is at the bottom of its motion.

What is the interval between successive times when the acceleration of P is equal and opposite to the acceleration of Q?

A. $\frac{T}{4}$

B. $\frac{T}{2}$

C. $\frac{3T}{4}$

D. T

Object P moves vertically with simple harmonic motion (shm). Object Q moves in a vertical circle with a uniform speed. P and Q have the same time period T. When P is at the top of its motion, Q is at the bottom of its motion.

What is the interval between successive times when the acceleration of P is equal and opposite to the acceleration of Q?

A. $\frac{T}{4}$

B. $\frac{T}{2}$

C. $\frac{3T}{4}$

D. T

An object undergoes simple harmonic motion (shm) of amplitude $x$₀. When the displacement of the object is $\frac{x_0}{3}$, the speed of the object is $v$. What is the speed when the displacement is $x$₀?

A. 0

B. $\frac{v}{3}$

C. $\frac{\sqrt{2}}{3}v$

D. $3v$

An object undergoes simple harmonic motion (shm) of amplitude $x$₀. When the displacement of the object is $\frac{x_0}{3}$, the speed of the object is $v$. What is the speed when the displacement is $x$₀?

A. 0

B. $\frac{v}{3}$

C. $\frac{\sqrt{2}}{3}v$

D. $3v$

A simple pendulum and a mass–spring system oscillate with the same time period. The mass of the pendulum bob and the mass on the spring are initially identical. The masses are halved.

What is $\frac{\mathrm{time} \mathrm{period} \mathrm{of} \mathrm{pendulum}}{\mathrm{time} \mathrm{period} \mathrm{of} \mathrm{mass}–\mathrm{spring} \mathrm{system}}$ when the masses have been changed?

A.  $\frac{\sqrt{2}}{2}$

B.  $1$

C.  $\sqrt{2}$

D.  $2$

A simple pendulum and a mass–spring system oscillate with the same time period. The mass of the pendulum bob and the mass on the spring are initially identical. The masses are halved.

What is $\frac{\mathrm{time} \mathrm{period} \mathrm{of} \mathrm{pendulum}}{\mathrm{time} \mathrm{period} \mathrm{of} \mathrm{mass}–\mathrm{spring} \mathrm{system}}$ when the masses have been changed?

A.  $\frac{\sqrt{2}}{2}$

B.  $1$

C.  $\sqrt{2}$

D.  $2$

Outline why the cylinder performs simple harmonic motion when released.

Outline why the cylinder performs simple harmonic motion when released.

Outline why the cylinder performs simple harmonic motion when released.

The mass of the cylinder is $118 \mathrm{kg}$ and the cross-sectional area of the cylinder is $2.29\times {10}^{-1} {\mathrm{m}}^2$. The density of water is $1.03\times {10}^3 \mathrm{kg} {\mathrm{m}}^{-3}$. Show that the angular frequency of oscillation of the cylinder is about $4.4 \mathrm{rad} {\mathrm{s}}^{-1}$.

The mass of the cylinder is $118 \mathrm{kg}$ and the cross-sectional area of the cylinder is $2.29\times {10}^{-1} {\mathrm{m}}^2$. The density of water is $1.03\times {10}^3 \mathrm{kg} {\mathrm{m}}^{-3}$. Show that the angular frequency of oscillation of the cylinder is about $4.4 \mathrm{rad} {\mathrm{s}}^{-1}$.

The mass of the cylinder is $118 \mathrm{kg}$ and the cross-sectional area of the cylinder is $2.29\times {10}^{-1} {\mathrm{m}}^2$. The density of water is $1.03\times {10}^3 \mathrm{kg} {\mathrm{m}}^{-3}$. Show that the angular frequency of oscillation of the cylinder is about $4.4 \mathrm{rad} {\mathrm{s}}^{-1}$.

Determine the maximum kinetic energy $E_{\mathrm{kmax}}$ of the cylinder.

Determine the maximum kinetic energy $E_{\mathrm{kmax}}$ of the cylinder.

Determine the maximum kinetic energy $E_{\mathrm{kmax}}$ of the cylinder.

Draw, on the axes, the graph to show how the kinetic energy of the cylinder varies with time during one period of oscillation $T$.

Draw, on the axes, the graph to show how the kinetic energy of the cylinder varies with time during one period of oscillation $T$.

Draw, on the axes, the graph to show how the kinetic energy of the cylinder varies with time during one period of oscillation $T$.

A mass–spring system oscillates vertically with a period of $T$ at the surface of the Earth. The gravitational field strength at the surface of Mars is $0.3 \text{g}$. What is the period of the same mass–spring system on the surface of Mars?

A.  $0.9T$

B.  $0.3T$

C.  $T$

D.  $3T$

A mass–spring system oscillates vertically with a period of $T$ at the surface of the Earth. The gravitational field strength at the surface of Mars is $0.3 \text{g}$. What is the period of the same mass–spring system on the surface of Mars?

A.  $0.9T$

B.  $0.3T$

C.  $T$

D.  $3T$

An object performs simple harmonic motion (shm). The graph shows how the velocity v of the object varies with time t.

The displacement of the object is x and its acceleration is a. What is the variation of x with t and the variation of a with t?

An object performs simple harmonic motion (shm). The graph shows how the velocity v of the object varies with time t.

The displacement of the object is x and its acceleration is a. What is the variation of x with t and the variation of a with t?

The bob of a pendulum has an initial displacement $x_0$ to the right. The bob is released and allowed to oscillate. The graph shows how the displacement varies with time. At which point is the velocity of the bob at its maximum magnitude directed towards the left?

The bob of a pendulum has an initial displacement $x_0$ to the right. The bob is released and allowed to oscillate. The graph shows how the displacement varies with time. At which point is the velocity of the bob at its maximum magnitude directed towards the left?

Calculate, in m s⁻¹, the maximum velocity of vibration of point P when it is vibrating with a frequency of 195 Hz.

Calculate, in m s⁻¹, the maximum velocity of vibration of point P when it is vibrating with a frequency of 195 Hz.

Calculate, in m s⁻¹, the maximum velocity of vibration of point P when it is vibrating with a frequency of 195 Hz.

Calculate, in terms of g, the maximum acceleration of P.

Calculate, in terms of g, the maximum acceleration of P.

Calculate, in terms of g, the maximum acceleration of P.

Estimate the displacement needed to double the energy of the string.

Estimate the displacement needed to double the energy of the string.

Estimate the displacement needed to double the energy of the string.

A particle undergoes simple harmonic motion of amplitude $x_0$ and frequency $f$. What is the average speed of the particle during one oscillation?

A.  $0$

B.  $f x_0$

C.  $2f x_0$

D.  $4f x_0$

A particle undergoes simple harmonic motion of amplitude $x_0$ and frequency $f$. What is the average speed of the particle during one oscillation?

A.  $0$

B.  $f x_0$

C.  $2f x_0$

D.  $4f x_0$

Calculate the period of oscillations of q.

Calculate the period of oscillations of q.

Calculate the period of oscillations of q.

A plate performs simple harmonic oscillations with a frequency of 39 Hz and an amplitude of 8.0 cm.

Show that the maximum speed of the oscillating plate is about 20 m s⁻¹.

A plate performs simple harmonic oscillations with a frequency of 39 Hz and an amplitude of 8.0 cm.

Show that the maximum speed of the oscillating plate is about 20 m s⁻¹.

A plate performs simple harmonic oscillations with a frequency of 39 Hz and an amplitude of 8.0 cm.

Show that the maximum speed of the oscillating plate is about 20 m s⁻¹.

A simple pendulum has a time period $T$ on the Earth. The pendulum is taken to the Moon where the gravitational field strength is $\frac{1}{6}$ that of the Earth.

What is the time period of the pendulum on the Moon?

A.  $T\sqrt{6}$

B.  $T$

C.  $\frac{\sqrt{6}}{6}T$

D.  $\frac{T}{6}$

A simple pendulum has a time period $T$ on the Earth. The pendulum is taken to the Moon where the gravitational field strength is $\frac{1}{6}$ that of the Earth.

What is the time period of the pendulum on the Moon?

A.  $T\sqrt{6}$

B.  $T$

C.  $\frac{\sqrt{6}}{6}T$

D.  $\frac{T}{6}$

Outline two reasons why both models predict that the motion is simple harmonic when $a$ is small.

Outline two reasons why both models predict that the motion is simple harmonic when $a$ is small.

Outline two reasons why both models predict that the motion is simple harmonic when $a$ is small.

Determine the time period of the system when $a$ is small.

Determine the time period of the system when $a$ is small.

Determine the time period of the system when $a$ is small.

Outline, without calculation, the change to the time period of the system for the model represented by graph B when $a$ is large.

Outline, without calculation, the change to the time period of the system for the model represented by graph B when $a$ is large.

Outline, without calculation, the change to the time period of the system for the model represented by graph B when $a$ is large.

The graph shows for model A the variation with $x$ of elastic potential energy E_p stored in the spring.

Describe the graph for model B.

The graph shows for model A the variation with $x$ of elastic potential energy E_p stored in the spring.

Describe the graph for model B.

The graph shows for model A the variation with $x$ of elastic potential energy E_p stored in the spring.

Describe the graph for model B.

Which graph represents the variation with displacement of the potential energy P and the total energy T of a system undergoing simple harmonic motion (SHM)?

Which graph represents the variation with displacement of the potential energy P and the total energy T of a system undergoing simple harmonic motion (SHM)?

Which graph represents the variation with displacement of the potential energy P and the total energy T of a system undergoing simple harmonic motion (SHM)?

Which graph represents the variation with displacement of the potential energy P and the total energy T of a system undergoing simple harmonic motion (SHM)?

A simple pendulum oscillates with frequency $f$. The length of the pendulum is halved. What is the new frequency of the pendulum?

A.  $2f$

B.  $\sqrt{2}f$

C.  $\frac{f}{\sqrt{2}}$

D.  $\frac{f}{2}$

A simple pendulum oscillates with frequency $f$. The length of the pendulum is halved. What is the new frequency of the pendulum?

A.  $2f$

B.  $\sqrt{2}f$

C.  $\frac{f}{\sqrt{2}}$

D.  $\frac{f}{2}$

A simple pendulum oscillates with frequency $f$. The length of the pendulum is halved. What is the new frequency of the pendulum?

A.  $2f$

B.  $\sqrt{2}f$

C.  $\frac{f}{\sqrt{2}}$

D.  $\frac{f}{2}$

A simple pendulum oscillates with frequency $f$. The length of the pendulum is halved. What is the new frequency of the pendulum?

A.  $2f$

B.  $\sqrt{2}f$

C.  $\frac{f}{\sqrt{2}}$

D.  $\frac{f}{2}$

A mass oscillating in simple harmonic motion on the end of a spring has an amplitude $x$₀ and a total energy E_T. The mass on the spring is doubled and made to oscillate with the same amplitude $x$₀.

What is the total energy of the oscillating system after the change?

A.  E_T

B.  $\sqrt{2}$E_T

C.  2E_T

D.  4E_T

A mass oscillating in simple harmonic motion on the end of a spring has an amplitude $x$₀ and a total energy E_T. The mass on the spring is doubled and made to oscillate with the same amplitude $x$₀.

What is the total energy of the oscillating system after the change?

A.  E_T

B.  $\sqrt{2}$E_T

C.  2E_T

D.  4E_T

A mass oscillating in simple harmonic motion on the end of a spring has an amplitude $x$₀ and a total energy E_T. The mass on the spring is doubled and made to oscillate with the same amplitude $x$₀.

What is the total energy of the oscillating system after the change?

A.  E_T

B.  $\sqrt{2}$E_T

C.  2E_T

D.  4E_T

A mass oscillating in simple harmonic motion on the end of a spring has an amplitude $x$₀ and a total energy E_T. The mass on the spring is doubled and made to oscillate with the same amplitude $x$₀.

What is the total energy of the oscillating system after the change?

A.  E_T

B.  $\sqrt{2}$E_T

C.  2E_T

D.  4E_T

Show that the phase difference between the oscillations of the two corks is $\pi$ radians.

Show that the phase difference between the oscillations of the two corks is $\pi$ radians.

Show that the phase difference between the oscillations of the two corks is $\pi$ radians.

The magnitude of the net force on q is given by $\frac{32kQq}{D^3}x$. Explain why the charge q will execute simple harmonic oscillations about C.

The magnitude of the net force on q is given by $\frac{32kQq}{D^3}x$. Explain why the charge q will execute simple harmonic oscillations about C.

The magnitude of the net force on q is given by $\frac{32kQq}{D^3}x$. Explain why the charge q will execute simple harmonic oscillations about C.

The mass of the charge q is 0.025 kg.

Calculate the angular frequency of the oscillations using the data in (a)(ii) and the expression in (b)(i).

The mass of the charge q is 0.025 kg.

Calculate the angular frequency of the oscillations using the data in (a)(ii) and the expression in (b)(i).

The mass of the charge q is 0.025 kg.

Calculate the angular frequency of the oscillations using the data in (a)(ii) and the expression in (b)(i).

The charges Q are replaced by neutral masses M and the charge q by a neutral mass m. The mass m is displaced away from C by a small distance $x$ and released. Discuss whether the motion of m will be the same as that of q.

The charges Q are replaced by neutral masses M and the charge q by a neutral mass m. The mass m is displaced away from C by a small distance $x$ and released. Discuss whether the motion of m will be the same as that of q.

The charges Q are replaced by neutral masses M and the charge q by a neutral mass m. The mass m is displaced away from C by a small distance $x$ and released. Discuss whether the motion of m will be the same as that of q.

Sketch the phase difference between the oscillations of the two corks is $\pi$ radians.

Sketch the phase difference between the oscillations of the two corks is $\pi$ radians.

Sketch the phase difference between the oscillations of the two corks is $\pi$ radians.