Simple Harmonic Oscillations (CIE A Level Physics)

A mass of 0.42 kg is attached to a spring and the system is made to oscillate with simple harmonic motion (SHM) on a horizontal, frictionless surface. The mass passes through the equilibrium position 200 times per minute.  

The kinetic energy of the mass as it passes through the equilibrium position is 500 mJ. There are two points where the restoring force acting on the mass is at its maximum. 

Show that the distance between these points is approximately 29 cm.

Sketch a graph to show how the velocity of the mass varies with time. Label the graph with any suitable values.

Find the distance of the mass from the equilibrium position when the speed of the block is 0.8 m s^–1

The experiment is moved to planet X. The gravitational acceleration on planet X is g. It is known that  ​​ = 2. Another change is that three more identical springs are placed in parallel to the original spring. 

The period of a spring undergoing simple harmonic oscillation is given by:

where m  is the mass of the object at the end of the spring and k  is the spring constant. 

Describe, without calculations, how these changes affect the frequency with which the mass oscillates.

A small wooden cuboid block of mass m floats in water, as shown in Fig. 1.1.

The top face of the block is horizontal and has an area A. The density of the water is ρ.

The block is displaced downwards as shown in Fig. 1.2 so that the surface of the water is now higher up the block.

State and explain the direction of the resultant force acting on the wooden block in this position.

The block in (a) is now released so that is oscillates vertically. 

The resultant force F acting on the block is given by 

where g is the gravitational field strength and x is the vertical displacement of the block from the equilibrium position.  

Explain why the oscillations of the block are simple harmonic. 

Use the equation from (b) to obtain an expression for the angular frequency ω of the oscillations of the block.

The block is now placed in a liquid with a lower density. The block is displaced and released so that it oscillates vertically. The variation with displacement x of the acceleration a of the block is measured for the first half of the oscillation, as shown in Fig. 1.3. 

Use Fig. 1.3. to determine the decrease  in energy of the oscillation for the first half of the oscillation.