Simple Harmonic Motion (DP IB Physics: HL): Exam Questions

A pendulum bob is suspended from a fixed point by a string of negligible mass. The bob is displaced a horizontal distance $x_0$ from its rest position and then released.

Explain why the pendulum will perform simple harmonic oscillations when the bob is released.

Sketch the variation with time of the displacement for one complete swing of the pendulum. Label the peak value on the displacement axis.

The pendulum completes 10 oscillations in 12.0 s.

Calculate the frequency of the oscillations.

The pendulum is shortened to a quarter of its original length.

Determine the effect this has on the frequency of the oscillations.

A ball suspended by a string from a fixed support oscillates with simple harmonic motion. The amplitude of the oscillation is 5 cm, and the maximum acceleration is 2.0 m s^-2.

Sketch, on the axes below, a graph to show how the acceleration of the ball varies with its displacement from the rest position.

Using your graph from (a), determine the period $T$ of the oscillation.

At time $t=0$, the ball is released from position X and allowed to oscillate. Positive displacements mean displacements to the right of the equilibrium position.

State and explain the direction of the velocity of the ball at time $t=\frac{T}{8}$.

Outline the energy transfers occurring as the ball on the string completes half an oscillation, starting from position X.

A smooth glass marble is held at the edge of a bowl and released. The marble rolls up and down the sides of the bowl with simple harmonic motion.

The magnitude of the restoring force which returns the marble to equilibrium is given by:

 $F$$=\frac{mgx}{R}$

  Where $x$ is the displacement at a given time, and $R$ is the radius of the bowl.

Outline why the oscillations can be described as simple harmonic motion.

Describe the energy changes that take place during the simple harmonic motion of the marble.

As the marble is released it has potential energy of 15 μJ. The mass of the marble is 3 g.

Calculate the velocity of the marble at the equilibrium position.

Sketch a graph to represent the kinetic, potential and total energy of the motion of the marble, assuming no energy is dissipated as heat. Label any relevant values on the graph.

A block attached to the end of a horizontal spring is initially at rest on a frictionless surface, as shown. The block is held at position 1 and released so that it oscillates with a simple harmonic motion between positions 1 and 3.

Sketch, on the axes provided, a graph to show how the acceleration of the block varies with its displacement from equilibrium.

The graph shows the variation of velocity $v$ with time $t$ for the block during one cycle of the oscillation.

A point on the graph has been labelled that represents a point Y located between positions 1 and 3.

(i) Sketch, on the axes, the variation of displacement $x$ with time $t$ for the block during one cycle of the oscillation.

[2]

(ii) Label on your graph the points that represent positions 1, 2 and 3.

[1]

(iii) State the direction of the displacement of the block at Y.

[1]

At point Y, the kinetic energy of the block and the potential energy stored in the spring are equal.

Explain how many times this occurs during one oscillation.

An experiment is carried out on planet Z using a simple pendulum and a mass-spring system. The pendulum has a length $l$, and the mass-spring system consists of a block of mass $m$ attached to a spring with spring constant $k$.

The block moves horizontally on a frictionless surface. A motion sensor with a lightbulb is placed above the equilibrium position of the block. Each time the block passes the equilibrium position, the lightbulb lights up.

The pendulum and the block are both displaced from their equilibrium positions and oscillate with simple harmonic motion. The pendulum completes 150 full oscillations in seven minutes and the bulb lights up once every 0.70 seconds. The block has a mass of 349 g.

Show that $k$ is approximately 7 N m⁻¹. 

The volume of planet Z is the same as the volume of Earth, and the density of planet Z is twice the density of Earth.

(i) Show that $l=\frac{4mg}{k}$

[2]

(ii) Calculate the value of $l$.

[2]

Discuss whether conducting the experiment on Planet Z, rather than on Earth, has an effect on the period of the pendulum and the mass-spring system.

A solid vertical cylinder of mass $m$ and uniform cross-sectional area $A$ floats, partially submerged, in water. When the cylinder is floating at rest, a mark is aligned with the surface of the water. The cylinder is pushed vertically downwards so that the mark is a distance $x$ below the water surface.

The cylinder is released at time $t=0$. The resultant vertical force $F$ on the cylinder is related to the displacement $x$ of the mark by

$F=-\rho Agx$

where $\rho$ is the density of water.

Outline why the cylinder performs simple harmonic motion when released.

Show that the angular frequency $\omega$ of oscillation of the cylinder is

$\omega =\sqrt{\frac{\rho Ag}{m}}$

The following data are available.

• Mass of cylinder = 95.0 kg

• Cross-sectional area of cylinder = 1.22 × 10^-2 m²

• Density of water = 1.03 × 10³ kg m^-3

• Displacement of cylinder below water surface = 0.550 m

Calculate

(i) the angular frequency $\omega$ of oscillation of the cylinder

[1]

(ii) the maximum kinetic energy $E_{kmax}$ of the cylinder.

[2]

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

A ball-spring system oscillates horizontally about an equilibrium point.

Describe the conditions required for the ball-spring system to perform simple harmonic motion.

The ball begins oscillating from the equilibrium position. The graph shows how the displacement of the ball varies with time.

Determine the maximum velocity of the ball.

Using the graph in part (b):

(i) Show that the acceleration at 90 ms is 43 m s⁻².

[2]

(ii) Label a letter X on the graph at a point where the resultant force acting on the ball is zero.

[1]

A steel block with a mass of 45 g on a spring undergoes simple harmonic motion with a period of 0.84 s.

The steel block is replaced with a wooden block. The wooden block attached to the same spring undergoes simple harmonic motion with a period of 0.64 s. Both blocks are displaced horizontally by 3.6 cm from the equilibrium position on a frictionless surface. 

Determine the total energy in the oscillation of the wooden block.

Sketch, on the axes, the variation with displacement of

• the total energy of the wooden block and spring

• the kinetic energy of the wooden block

• the potential energy stored in the spring

Label any relevant values on the axes.

In a separate experiment, a 45 g mass attached to a spring is set to oscillate in simple harmonic motion.

The graph shows the variation with time of the displacement of the mass on the spring.

(i) Describe the motion of the mass on the spring.

[1]

(ii) Determine the initial potential energy of the mass-spring system.

[2]

A mass of 75 g is connected between two identical springs. The mass-spring system rests on a frictionless surface. A force of 0.025 N is needed to compress or extend the spring by 1.0 mm. 

The mass is pulled from its equilibrium position to the right by 0.055 m and then released. The mass oscillates about the equilibrium position in simple harmonic motion.

The mass-spring system can be used to model the motion of an ion in a crystal lattice structure. 

The frequency of the oscillation of the ion is 8 × 10¹² Hz and the mass of the ion is 6 × 10⁻²⁶ kg. The amplitude of the vibration of the ion is 2 × 10⁻¹¹ m.

For the oscillations of the mass-spring system:

(i) Calculate the acceleration of the mass at the moment of release

[2]

(ii) Estimate the maximum kinetic energy of the ion.

[2]

For the mass-spring system 

(i) Calculate the total energy of the system.

[1]

(ii) Use the axes to sketch a graph showing the variation over time of the kinetic energy of the mass and the potential energy of the springs. 

You should include appropriate values, and show the oscillation over one full period.

[2]

The same mass and a single spring from part (a) are attached to a rigid horizontal support. 

The length of the spring with the mass attached is 64 mm. The mass is pulled downwards until the length of the spring is 76 mm. The mass is released, and the vertical mass-spring system performs simple harmonic motion.

For the new mass-spring system, determine

(i) the velocity of the mass 2 seconds after its release. 

[1]

(ii) the kinetic energy of the mass at this point.

[2]

The diagram shows the vertical spring-mass system as it moves through one period.

Label the diagram to show when:

• E_p = max

• E_k = max

• v = 0

• v = max

A ball is displaced through a small distance $x$ from the bottom of a bowl and is then released from rest. 

The frequency of the resulting oscillation is 1.5 Hz and the maximum velocity reaches 0.36 m s⁻¹. The radius of the bowl is $r$.

For the oscillating ball: 

(i) Show that $r$ is approximately 11 cm.

[2]

(ii) Calculate the amplitude of oscillation.

[2]

Sketch, on the axes, graphs to show the variation with time of the displacement, velocity and acceleration of the ball during one period

Include any relevant values on the axes.

The ball was replaced by a ball of the same size, but with a greater mass. 

Outline the effect this would have on the period of the oscillation.

A mass-spring system has been set up horizontally on the lab bench, so that the mass can oscillate.

The time period of the mass is given by the equation:

$T=2\mathrm{\pi }\sqrt{\frac{m}{k}}$

(i) Calculate the spring constant of a spring attached to a mass of 0.7 kg and time period 1.4 s.

 [1]

(ii) Outline the condition under which the equation can be applied.

[1]

Sketch a velocity-displacement graph of the motion of the block as it undergoes simple harmonic motion.

A new mass of m = 50 g replaces the 0.7 kg mass and is now attached to the mass-spring system.

The graph shows the variation with time of the velocity of the block.

Determine the total energy of the system with this new mass.

Determine the potential energy of the system after 6 seconds have passed.

A volume of water in a U-shaped tube performs simple harmonic motion.

State and explain the phase difference between the displacement and the acceleration of the upper surface of the water.

The U-tube is tipped and then set upright, to start the water oscillating. Over a period of a few minutes, a motion sensor attached to a data logger records the change in velocity from the moment the U-tube is tipped. Assume there is no friction in the tube.

Sketch the graph the data logger would produce.

The cross-sectional area of the tube is $A$. The height difference between the two arms of the tube is $h$, and the density of the water is $\rho$.

Show that the restoring force for the motion is given by $F=\rho ghA$.

The time period of the oscillating water is given by $T=2\pi \sqrt{\frac{L}{g}}$ where $L$ is the height of the water column at equilibrium and g is the acceleration due to gravity.

If $L$ is 15 cm, determine the frequency of the oscillations.

The diagram shows a flat metal disk placed horizontally, that oscillates in the vertical plane.

The graph shows how the disk's acceleration, a, varies with displacement, x.

Show that the oscillations of the disk are an example of simple harmonic motion.

Some grains of salt are placed onto the disk. 

The amplitude of the oscillation is increased gradually from zero.

At amplitude A_Z, the grains of salt are seen to lose contact with the metal disk. 

(i) Determine and explain the acceleration of the disk when the grains of salt first lose contact with it.

[3]

(ii) Deduce the value of amplitude A_Z.

[1]

A group of students construct a model of the Moon orbiting the Earth to demonstrate the phases of the Moon. 

The model is built on a turntable of radius r that rotates uniformly with an angular speed ω. The students use LED lights to represent parallel beams of incident light from the Sun. 

The diagram shows the model as viewed from above.

The students observe the shadow of the model Moon on a wall.

At time t = 0, θ = 0 and the shadow of the model Moon could not be seen at position E as it passed through the shadow of the model Earth. 

Some time later, the shadow of the model Moon could be seen at position X.

(i) Show that the distance EX is equivalent to $r\sin \omega t$.

[2]

(ii) Describe the motion of the shadow of the Moon on the wall.

[1]

The diameter of the turntable is d = 50 cm, and it rotates with an angular speed of ω = 2.3 rad s⁻¹.

For the motion of the shadow of the model Moon, calculate: 

(i) the amplitude, A.

[1]

(ii) the period, T.

[1]

(iii) the speed at which the shadow passes through position E.

[2]

The defining equation of SHM links acceleration, a, angular speed, ω, and displacement, x.

 $a=-{\omega }^{2}x$

For the shadow of the model Moon: 

(i) Determine the magnitude of the acceleration when the shadow is instantaneously at rest.

[2]

(ii) Deduce the change in the maximum acceleration if the angular speed were reduced by a factor of 4 and the diameter of the turntable were reduced by half.

[1]

The needle carrier of a sewing machine moves with simple harmonic motion. The needle carrier is constrained to move on a vertical line by low friction guides, whilst the disk and peg rotate in a circle. As the disk completes one oscillation, the needle completes one stitch.

The sewing machine completes 840 stitches in one minute.

Calculate the angular speed of the peg.

The needle carrier has a mass of 23.9 g, and the needle has a mass of 0.7 g. The needle moves a distance of 36 mm between its extremities of movement.

Assuming that the fabric requires a negligible force for the needle to penetrate it: 

(i) Calculate the maximum speed of the needle.

[1]

(ii) Determine the kinetic energy of the needle carrier system,at this point.

[1]

For the needle-carrier system: 

(i) Label, on the diagram, the positions of the peg at the point of maximum velocity, and the points of maximum contact force of the peg on the slot.

[2]

(ii) Calculate the maximum force acting on the peg by the slot.

[1]

A metal pendulum bob of mass 7.5 g is suspended from a fixed point by a length of thread with negligible mass. The pendulum is set in motion and oscillates with simple harmonic motion. 

The graph shows the kinetic energy of the bob as a function of time.

Calculate the length of the thread.

For the simple pendulum: 

(i) Label the graph from part a with an A at the point where the restoring force is acting at a maximum.

[1]

(ii) Label the graph from part a with a B at the point where the speed of the pendulum is half of its initial speed.

[1]

Show that the amplitude of the oscillation is around 0.6 m.

A steel spring with an unstretched length of 33 cm is attached to a fixed point and a mass of 35 g is attached and gently lowered until equilibrium is reached and the spring has a length of 37.5 cm. The spring is then stretched elastically to a length of 42 cm and released. 

Design a plan to investigate if the oscillation is simple harmonic motion.

For the stretching of the spring: 

(i) Calculate the gravitational potential energy lost by the mass.

[1]

(ii) Determine the elastic potential energy gained by the spring.

[2]

(iii) Explain why the two answers are different.

[1]

For the simple harmonic oscillation: 

(i) Determine the resultant force acting on the load at the lowest point of its movement.

[2]

(ii) Calculate the maximum speed of the mass.

[2]

A student with mass 68 kg hangs from a bungee cord with a spring constant, k = 270 N m⁻¹. The student is pulled down to a point where the cord is 4.0 m longer than its unstretched length, and then released. The student oscillates with SHM.

For the student: 

(i) Determine their position 15.7 s after being released.

[3]

(ii) Calculate their velocity 15.7 s after being released.

[1]

(iii) Explain where in the oscillation the student is at 15.7 s after being released. You may want to include a sketch diagram to aid your explanation.

[2]

A second student wants to do the bungee jump, however, they would like a greater number of bounces in their five minute session. 

Evaluate the possibilities for facilitating the student's wishes.