9.1 – Simple harmonic motion

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 $4.4 \mathrm{rad} {\mathrm{s}}^{-1}$.

Estimate the value of k in the following expression.

T = $2\pi \sqrt{\frac{m}{k}}$

Give an appropriate unit for your answer. Ignore the mass of the cable and any oscillation of satellite X.

Estimate the value of k in the following expression.

T = $2\pi \sqrt{\frac{m}{k}}$

Give an appropriate unit for your answer. Ignore the mass of the cable and any oscillation of satellite X.

Estimate the value of k in the following expression.

T = $2\pi \sqrt{\frac{m}{k}}$

Give an appropriate unit for your answer. Ignore the mass of the cable and any oscillation of satellite X.

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.

Calculate the speed of the block as it passes the equilibrium position. 

Calculate the speed of the block as it passes the equilibrium position. 

Calculate the speed of the block as it passes the equilibrium position. 

determine the initial energy.

determine the initial energy.

determine the initial energy.

Show that the period of oscillation of the ball is about 6 s.

Show that the period of oscillation of the ball is about 6 s.

Show that the period of oscillation of the ball is about 6 s.

Calculate the period of oscillations of q.

Calculate the period of oscillations of q.

Calculate the period of oscillations of q.

An object undergoing simple harmonic motion (SHM) has a period T and total energy E. The amplitude of oscillations is halved. What are the new period and total energy of the system?

An object undergoing simple harmonic motion (SHM) has a period T and total energy E. The amplitude of oscillations is halved. What are the new period and total energy of the system?

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.

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.  $\frac{T}{6}$

Determine the amplitude of oscillation for test 1.

Determine the amplitude of oscillation for test 1.

Determine the amplitude of oscillation for test 1.

Deduce $\frac{k_1}{k_2}$.

Deduce $\frac{k_1}{k_2}$.

Deduce $\frac{k_1}{k_2}$.

Label on the graph with the letter X a point where the speed of the pendulum is half that of its initial speed.

Label on the graph with the letter X a point where the speed of the pendulum is half that of its initial speed.

Label on the graph with the letter X a point where the speed of the pendulum is half that of its initial speed.

The mass of the pendulum bob is 75 g. Show that the maximum speed of the bob is about 0.7 m s^–1.

The mass of the pendulum bob is 75 g. Show that the maximum speed of the bob is about 0.7 m s^–1.

The mass of the pendulum bob is 75 g. Show that the maximum speed of the bob is about 0.7 m s^–1.

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. $3v$

Calculate the time taken for the block to return to the equilibrium position for the first time. 

Calculate the time taken for the block to return to the equilibrium position for the first time. 

Calculate the time taken for the block to return to the equilibrium position for the first time. 

The amplitude of oscillation is 0.12 m. On the axes, draw a graph to show the variation with time t of the velocity v of the ball during one period.

The amplitude of oscillation is 0.12 m. On the axes, draw a graph to show the variation with time t of the velocity v of the ball during one period.

The amplitude of oscillation is 0.12 m. On the axes, draw a graph to show the variation with time t of the velocity v of the ball during one 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. $\frac{2\pi A}{T}\cos \left(\frac{\pi t}{T}\right)$ 

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?

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.  $2$

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 $T$.

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.

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.  $3T$

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.

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.

Deduce whether the motion of Z is simple harmonic.

Deduce whether the motion of Z is simple harmonic.

Deduce whether the motion of Z is simple harmonic.

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 standing wave with a first harmonic of frequency $f_1$ is formed on a string fixed at both ends.

The frequency of the third harmonic is $f_3$.

What is $\frac{f_1}{f_3}$?

A.  3

B.  $\frac{3}{2}$

C.  $\frac{2}{3}$

D.  $\frac{1}{3}$

A standing wave with a first harmonic of frequency $f_1$ is formed on a string fixed at both ends.

The frequency of the third harmonic is $f_3$.

What is $\frac{f_1}{f_3}$?

A.  3

B.  $\frac{3}{2}$

C.  $\frac{2}{3}$

D.  $\frac{1}{3}$

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

Explain the pattern seen on the screen.

Explain the pattern seen on the screen.

Explain the pattern seen on the screen.