AH Simple Harmonic Motion Questions

1. a) State what is meant by the term ‘Simple Harmonic Motion’. b) Using the equation F = - ky, state the quantities and units involved. c) Using F = - ky and Newton’s second law of motion, show that d2y/dt2 = - (k/m)y. 2. The equation used for SHM at y = 0 and t = 0 is given by y = A sinωt.

a) Find the equation for the velocity of the SHM using differentiation.

b) How is the equation found in a) adapted to find the maximum velocity and at which part of the oscillation is it found? c) Find the equation for the acceleration of the SHM using differentiation. d) How is the equation found in c) adapted to find the maximum acceleration and at which part of the oscillation is this found? e) At which of the oscillation is the acceleration zero? 3. An object of mass 0.2kg is undergoing SHM of frequency 6Hz and amplitude 0.25m. Calculate or find:

a) Period of the oscillation.

b) Angular frequency.

c) Maximum speed.

d) Maximum force acting on the object.

4. a) State the equation for the kinetic energy of a particle undergoing SHM. b) Show that the equation for total energy of a particle undergoing SHM is ET = ½mω2A2. c) Show that the equation for kinetic energy of a particle undergoing SHM is Ep = ½mω2y2. d) An object of mass 0.35kg oscillates in SHM with an amplitude of 140mm and a frequency of 0.60Hz. Calculate or find: i) Maximum kinetic energy of the object. ii) Maximum potential energy of the object. iii) Potential and kinetic energy at the mid-way point between the centre and the extremity of the motion.

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10. A test tube contains lead shot. The combined mass of the test tube and the lead shot is 0.20kg. The test tube is gently dropped into a container of water and oscillates above and below its equilibrium position with simple harmonic motion as shown below.

The displacement y of the test tube from its equilibrium position is described by the equation

y = 0.06 cos8t, where y is in metres and t is in seconds.

a) Show that the kinetic energy of the test tube, in Joules, is given by the equation

EK = 6.4(3.6x10-3 – y2). b) Calculate the maximum value of the kinetic energy of the test tube.

c) Calculate the potential energy of the test tube when it is 50mm above its equilibrium position.

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12. A simple pendulum consists of a lead ball on the end of a long string as shown below.

The ball moves with simple harmonic motion. At time t the displacement of the ball is given by the expression

s = 4x10-2 cos3.6t where s is in metres and t is in seconds.

a) i) State the definition of simple harmonic motion. ii) Calculate the period of the pendulum.

b) Calculate the maximum speed of the ball.

c) The mass of the ball is 6x10-2kg and the string has negligible mass. Calculate the total energy of the pendulum.

d) The period of the pendulum is given by the expression

, where L is the length of the pendulum. Calculate the length of the pendulum.

e) In the above case, the asumption has been made that the amplitude of the motion is continually decreasing. What term is used to describe this?

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