Energy in Simple Harmonic Motion (CIE A Level Physics)

Determine the time period of the pendulum.

Label a point X on the graph where the pendulum is in equilibrium.

The mass of the pendulum bob is 60 × 10^–3 kg. 

i)   Determine the maximum kinetic energy of the pendulum bob.

ii)   Show that the maximum speed of the bob is about 0.82 m s^–1.

Fig. 1.1 shows a mass-spring system which undergoes simple harmonic motion.

The 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.

Fig. 1.1

Calculate the acceleration of the mass at the moment of release.

The mass-spring system from part (a) can be used to model the motion of an ion in a crystal lattice structure. The model is shown in Fig. 1.2.

Fig. 1.2

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.

Estimate the maximum kinetic energy of the ion.

For the mass-spring system in part (a) the total energy of the system is 0.077 J.

Sketch a graph showing the variation over time of the kinetic energy of the mass and the potential energy of the springs. 

You should use the axes given below, include appropriate values, and show the oscillation over one full period.

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

Fig 1.3 shows the vertical spring-mass system as it moves through one period.

Fig. 1.3

Annotate the diagram to show when:

Fig 1.1 shows a graph of the displacement over time of a simple pendulum oscillating in simple harmonic motion. 

Fig. 1.1

State the potential energy of the pendulum at points X, Y and Z in terms of maximum and minimum.

The amplitude of the pendulum is 60 cm when it swings with a frequency of 0.7 Hz. During this oscillation, the maximum kinetic energy of the pendulum is 23 J.

Determine the mass of the pendulum bob.

Calculate the kinetic energy of the pendulum when the pendulum bob is at a distance of from its point of maximum displacement.

A U shaped tube with cross-sectional area A contains a liquid of mass m, density ρ and volume V as shown in Fig. 1.1. When the water of length L is in equilibrium in the tube the levels of water on either side are equal. When the water is pushed down a distance x_max on one side of the tube then the water on the opposite side rises. Upon release the water undergoes simple harmonic motion as the water levels on each side rise and fall about the equilibrium position.

Fig. 1.1

Obtain an expression for the maximum kinetic energy of the liquid in terms of its cross-sectional area A, density ρ, maximum velocity v_max and length of water L. 

Obtain an expression in terms of cross-sectional area A, liquid density ρ, and x_max for the gravitational potential energy E_{p (max)} of the liquid in the U-shaped tube in Fig. 1.1.

Show that the time period of the oscillation of the liquid in the U shaped tube of Fig. 1.1 is given by the equation

Determine the length of the liquid in the pipe, L in centimetres and the corresponding angular velocity, ω when the ratio of maximum displacement and maximum velocity is 0.05 s in Fig. 1.1.

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

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

A mass of m = 50 g is attached to the mass-spring system.

The graph in Fig. 1.1 shows the variation with time of the velocity of the block.

Fig. 1.1

Determine the total energy of the system when it is oscillating with a mass of 50 g.

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

The mass-spring system is now fixed so the mass and spring oscillate vertically. The mass remains at 50 g and the angular velocity and time period the same as before.

Show that the total energy of the system is still the same as in part (b).