Simple Harmonic Motion (DP IB Physics: HL)

A pendulum undergoes small-angle oscillations.

(i) State the equation and each of the variables that define simple harmonic motion.

(ii) Explain the meaning of the equation 

Sketch a graph of displacement against time for one swing of the pendulum. Start the time at zero seconds when the pendulum is at the maximum positive displacement, , position.

The time period of 10 oscillations is found to be 12.0 s.

Determine the frequency when the bob is 1.0 cm from its equilibrium position.

The student wants to double the frequency of the pendulum swing. The time period, T of a simple pendulum is given by the equation:

                                                       T = 2π

where L is the length of the string and g is the acceleration due to gravity

Deduce the change which would achieve this.

State and explain whether the motion of the objects in graphs I, II and III are simple harmonic oscillations

Explain why, in practice, a freely oscillating pendulum cannot maintain a constant amplitude.

The motion of an object undergoing SHM is shown in the graph below.

For this oscillator, determine: 

(i) The amplitude, A.

[1]

(ii) The period, T.

[1]

(iii) The frequency, f.

[1]

Using the graph from part (c), state a time in seconds when the object performing SHM has: 

(i) Maximum positive velocity.

[1]

(ii) Maximum negative acceleration.

[1]

(iii) Maximum potential energy.

[1]

A ball of mass 44 g on a 25 cm string oscillating in simple harmonic motion obeys the following equation:

a = −ω²x

Show mathematically that the graph of this equation is a downward sloping straight line that goes through the origin.

The graph below shows the acceleration, a, as a function of displacement, x, of the ball on the string.

The angular speed, ω, in rad s⁻¹, is related to the frequency, f, of the oscillation by the following equation:

For the ball on the string, determine the period, T, of the oscillation.

The ball is held in position X and then let go. The ball oscillates in simple harmonic motion.

Explain the change in acceleration as the ball on the string moves through half an oscillation from position X.

You can assume the ball is moving at position X.

Describe the energy transfers occurring as the ball on the string completes half an oscillation 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:

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

Deduce mathematically why the oscillations of the marble are in 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. Clearly label any important values on the graph.

An object is attached to a light spring and set on a frictionless surface. It is allowed to oscillate horizontally. Position 2 shows the equilibrium point.

(i) Sketch a graph of acceleration against displacement for this motion.

[2]

(ii) On your graph, mark positions 1, 2 and 3 according to the diagram.

[1]

The mass begins its motion from position 1 and completes a full oscillation.

(i) Sketch a graph of velocity against time to show this.

[2]

(ii) On your graph, add labels to show points 1, 2 and 3

[2]

At the point marked Y on the graph, the potential energy of the block is E_P. The block has mass m, and the maximum velocity it achieves is v_max.

Determine an equation for the potential energy at the point marked X.

Give your answer in terms of v_max , and m.

The graph shows how the displacement x of the mass varies with time t.

Determine the frequency of the oscillations. 

A ball within a ball-spring system oscillates about an equilibrium point. 

Describe the conditions for simple harmonic motion.

The ball oscillates with simple harmonic motion. On the axes provided, taking the oscillation as beginning at the positive amplitude when t = 0, sketch for the ball's motion:  

(i) The variation of displacement with time, label this graph x

[1]

(ii) The variation of velocity with time, label this graph v

[1]

(iii) The variation of acceleration with time, label this graph a

[1]

In a separate experiment, the ball is oscillating from the equilibrium position. The graph shows the displacement of the ball over time.

Determine the maximum velocity of the ball.

Using the graph from part (c) for the motion of the ball 

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

[2]

(ii) On the graph, mark an X 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 removed. Now, a wooden block attached to the same spring undergoes simple harmonic motion with a period of 0.64 s. The wooden block is 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.

Using the information from part (a), sketch on the axes the kinetic, potential, and total energies of the oscillating wooden block as they vary with displacement.

In a separate experiment, another mass-spring system is set to oscillate in simple harmonic motion. The mass has a mass of 45 g.

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

For the new experiment on the mass-spring system 

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

[1]

(ii) Determine the initial 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

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

[1]

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

[2]

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

Annotate 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 then released. 

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

For the oscillating ball: 

(i) Show that the radius of the bowl in which it oscillates is approximately 11 cm.

[2]

(ii) Calculate the amplitude of oscillation.

[2]

Sketch the graphs showing how the displacement, velocity and acceleration of the ball vary with time. You should include any relevant values.

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

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

An experiment is carried out on Planet Z using a simple pendulum and a mass-spring system. The block moves horizontally on a frictionless surface. A motion sensor is placed above the equilibrium position of the block which lights up every time the block passes it. 

The pendulum and the block are displaced from their equilibrium positions and oscillate with simple harmonic motion. The pendulum bob 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 the value of the spring constant k is approximately 7 N m⁻¹. 

The volume of Planet Z is the same as the volume of Earth, but Planet Z is twice as dense. 

For the experiment on Planet Z

(i) Considering the motion of both the spring and the pendulum, show that the length of the pendulum,

[2]

(ii) Calculate the value of .

[2]

Compare and contrast how performing the experiment on Planet Z, rather than on Earth, affects the period of the oscillations of the pendulum and the mass-spring system.

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 = 2π

(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 when 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 height difference between the two arms of the tube is , and the density of the water is .

Construct an equation to find the restoring force, , for the motion and simplify as much as possible.

The time period of the oscillating water is given by  where  is the height of the water column at equilibrium and g is the acceleration due to gravity.

If  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]

For a homework project, some students constructed a model of the Moon orbiting the Earth to show the phases of the Moon. 

The model was built upon a turntable with radius r, that rotates uniformly with an angular speed ω.

The students positioned LED lights to provide parallel incident light that represented light from the Sun. 

The diagram shows the model as viewed from above.

The students noticed that the shadow of the model Moon could be seen on the 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

For this model Moon and Earth 

(i) Construct an expression for θ in terms of ω and t

[1]

(ii) Derive an expression for the distance EX in terms of r, ω and t

[1]

(iii) Describe the motion of the shadow of the Moon on the wall

[1]

The diameter, d, of the turntable is 50 cm and it rotates with an angular speed, ω, of 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 as the shadow passes through position E.

[2]

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

For the shadow of the model Moon: 

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

[2]

(ii) Without the use of a calculator, predict the change in the maximum acceleration  if the angular speed was reduced by a factor of 4 and the diameter of the turntable was half of its original length.

[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.