9.1 Simple Harmonic Motion

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:

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

An IB Physics class was discussing this experiment.

A student in the class said that increasing the mass of the pendulum bob would increase the period of the oscillation because increasing the mass would increase the inertia of the bob. 

The teacher said the student was incorrect. 

Discuss the teachers comments.

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:

For the simple harmonic oscillation:

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:

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.

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.

The investigation from part (a) is repeated. The same wooden block and spring are used, but a second identical spring is added in parallel. 

Suggest how this change will affect the fractional uncertainty in the mass of the wooden block. 

The spring from part (a) is attached to the mass in part (a), and oscillates freely in simple harmonic motion. 

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

For the new 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

[1]

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

[1]

For the mass-spring system

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

[2]

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

Annotate the diagram to show when:

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:

[1]

For the ball 

[1]

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)

Show that the length of the pendulum,

[2]

(ii)

Calculate the value of .

[2]

The angle that the pendulum string makes with the horizontal is 81.4 ° when the acceleration of the pendulum bob is at a maximum. 

Determine the maximum speed reached my the pendulum bob.

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.