Oscillations

Many motions repeat themselves at regular intervals, such as a swinging pendulum or a beating heart. Such motions are called periodic, and when a body moves to and fro about a mean position it is said to oscillate. This chapter introduces the period T, the frequency f = 1/T, displacement and phase. It concentrates on simple harmonic motion (SHM), where the displacement varies as x(t) = A cos(omega t + phi). Here A is the amplitude, omega the angular frequency and phi the phase constant. You will see that SHM is the projection of uniform circular motion on a diameter. You will derive the velocity v = -omega A sin(omega t + phi) and the acceleration a = -omega^2 x, and learn the force law F = -kx for SHM. The chapter explains how the kinetic and potential energies of an oscillator change during the motion while the total energy (1/2)kA^2 stays constant. It applies these ideas to a mass on a spring, with T = 2pi sqrt(m/k), and the simple pendulum, with T = 2pi sqrt(L/g).

Detailed Worksheet: Oscillations Section A - Definitions (10 marks) 1. Distinguish between periodic motion and oscillatory motion. (2 marks) 2. Define simple harmonic motion. (2 marks) 3. Write the expression for the period of a mass m attached to a spring of spring constant k. (2 marks) 4. What is a seconds pendulum? Find its length. Take g = 9.8 m/s^2 and pi^2 = 9.8. (2 marks) 5. At which positions are the speed and the acceleration of a particle in SHM maximum? (2 marks) Section B - Calculations and Applications (15 marks) 6. For the SHM x = 5 cos(2pi t + pi/4) cm, find the amplitude, angular frequency, period and initial phase. (3 marks) 7. A mass of 1 kg is attached to a spring of spring constant 100 N/m. Find the period of its oscillation. (3 marks) 8. A particle in SHM has amplitude 0.05 m and angular frequency 10 rad/s. Find its maximum speed and maximum acceleration. (3 marks) 9. Find the period of a simple pendulum of length 1 m. Take g = 9.8 m/s^2. (3 marks) 10. A spring oscillator has k = 100 N/m and amplitude 0.1 m. Find the total energy and the kinetic energy when the displacement is half the amplitude. (3 marks) Section C - Diagrams (10 marks) 11. Draw graphs of displacement, velocity and acceleration against time for a particle in SHM. (4 marks) 12. Draw a diagram showing SHM as the projection of uniform circular motion on a diameter. (3 marks) 13. Draw graphs of kinetic energy, potential energy and total energy against displacement for a particle in SHM. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. Show that the motion of a mass attached to a horizontal spring is simple harmonic and derive its period. (5 marks) 15. Derive the expression for the period of a simple pendulum for small oscillations. (5 marks) 16. A 3 kg mass is attached to a spring of spring constant 1200 N/m on a frictionless surface. It is pulled 2.0 cm from equilibrium and released. Find the frequency, the maximum acceleration and the maximum speed of the mass. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Take g = 9.8 m/s^2. Assume small oscillations for pendulums.