NYJC EJC 2026 Oscillations and Waves Tutorial

9814 H3 Physics (2026) Oscillations and Wave Motion Tutorial 1 Oscillations 1 (a) What are the conditions for simple harmonic motion? [2] (b) A point mass moves along the line AB (Fig. 1.1) with simple harmonic motion of amplitude 0.040 m and period 1.6 s. Fig. 1.1 At time t = 0, the mass is at the equilibrium position O, and is moving in the direction OA. What is the displacement of the mass at time t = 1.3 s? In which direction is it moving at this instant? [6] (c) An ideal spring of unstretched length L and force constant k is fastened between two fixed points X and Y a distance L apart just above a horizontal, frictionless table. A mass M is then clamped to the mid -point of the spring so that it rests on the table. The mass is drawn aside through a small distance x, perpendicular to the original line of the spring, and is then released, as shown in plan view in Fig. 1.2. Fig. 1.2 Find the initial acceleration of the mass, and hence determine whether the resulting oscillations are simple harmonic. (Hint: for ( )1, 1 1 n z z nz + = + ) [8] (d) Why is simple harmonic motion so important in Physics? [4] [N93/P1/Q4] 2 (a) A straight wire has unstretched length L and area of cross-section A. Find the relation between the force constant k of the wire and the Young modulus E of the metal of which it is made. Why does this relation not apply to a spring in the form of a coil of the same wire? [4] x

9814 H3 Physics (2026) Oscillations and Wave Motion Tutorial 2 (b) A mass m is supported by a spring of force constant k. A simple derivation shows that the period T of vertical oscillations of the mass is given by ( ) 12 2T m k= . What assumptions must be made if this expression is to hold? (You are not asked to derive the expression.) [3] (c) A mass M is supported by two springs, of force constants k1 and k2 respectively, connected in series as shown in Fig. 2.1. Fig. 2.1 Find the effective force constant of the spring combination, and hence deduce the period of vertical oscillation of the mass. [4] (d) The mass M is now set between the same two springs on a horizontal, frictionless surface, as shown in Fig. 2.2. Each spring is of unstretched length l. The distance between the clamps at each side is D, where D is greater than 2l. Fig. 2.2 (i) Find the distance d of the equilibrium position of the mass from the left - hand clamp. [3] (ii) The mass is then displaced a distance x to the right of the equilibrium position, along the axis of the springs. The distance x is sufficiently short for both springs to remain in tension. Show that, when released, the mass moves with simple harmonic motion. Find the period of the oscillations. [6] [N95/P1/Q7]

9814 H3 Physics (2026) Oscillations and Wave Motion Tutorial 3 3 (a) A particle of mass m moves in a straight line with simple harmonic motion. Its displacement x at time t is given by the expression sinx A t=  . (i) Write down an expression for the velocity v of the particle at time t, in terms of A,  and t. [1] (ii) Sketch a graph showing the variation with time t of the kinetic energy Ek of the particle. Label important features of your graph with corresponding values of Ek and t. [4] (iii) Use your graph in (ii) to show that the average value of the kinetic energy over one complete cycle of the motion is 221 4 kE mA  =  . Write down an expression for the average value pE of the potential energy of the particle due to its motion over one cycle. [2] (b) An electron of mass me and charge –e moves in a straight line with damped harmonic motion of angular frequen cy  . According to classical electromagnetic theory, the power P radiated by the electron at any instant is given by 22 3 06 eaP c=  , where a is the acceleration of the electron at that instant, 0 is the permittivity of free space, and c is the speed of electromagnetic waves in free space. It is the continuous radiation of energy by the electron that causes the motion to be damped. Because of the damping, the amplitude of the oscillation of the electron decreases exponentially with time. At time t, the amplitude A is given by 0 tA A e −=  where A0 and  are constants. (i) Write down, in terms of  and A, the average value of a2 (the square of the acceleration of the electron) over one complete cycle. Hence show that the average power P radiated over one cycle is given by 2 2 4 3 012 eAP c  =   . [2] (ii) Obtain an expression, in terms of A0,  , me and  , for the total energy Etot (kinetic energy plus potential energy) of the electron at time t. [2] (iii) Hence find the rate of change of Etot with time. Use your result to show that 22 3 012 e e mc=   . [3] (iv) Suppose that an oscillating electron were radiating light of wavelength 500 nm. Use the classical theory to calculate how long it would take for the intensity of the light to fall to half of its initial value. [6]

9814 H3 Physics (2026) Oscillations and Wave Motion Tutorial 4 Wave Motion 4 (a) Sound is propagated in air as a longitudinal progressive wave, in which there is a repeated sequence of displacements of the air particles. These displacements give rise to pressure variations; hence , the wave may also be described in terms of a repeated sequence of changes in pressure. (i) Describe the motion of the particles in a longitudinal wave. [2] (ii) Fig. 4.1 illustrates nine particles, equally spaced along the line AB, in still air. Fig. 4.1 A sound wave of wavelength equal to AB is sent th rough the air in the direction Ox. Copy Fig. 4.1 and underneath it draw a second diagram showing possible positions of the nine particles in the wave relative to the undisturbed positions they occupy in Fig. 4.1. Use information from this diagram to draw a graph to show how the displacement d of the particles varies with distance x along the direction of propagation. [2] (iii) Explain how the points of high and low pressure in the air can be deduced from your diagram of the positions of the particles in the wave. Hence draw a second graph to show how the pressure p in the air varies with x. State the phase difference, if any, between the displacement wave and the pressure wave. [5] (b) A characteristic of a progressive wave is that it transmits energy. (i) Define the intensity of a wave. [1] (ii) The intensity I of a sound wave in air is related to the amplitude do of the displacement of the air particles by 2 oAd=I , where A is a constant. Find A in terms of the speed of propagation c, the angular frequency  of the wave and the mean density o of the ai r. Assume the expression 2212totE ma=  for the total energy of a mass m moving in simple harmonic motion of amplitude a and angular frequency  . [4] (c) When a sound wave passes along a straight pipe of constant cross -section, energy is lost from the wave. The rate of loss of energy in each element x is proportional to the intensity of the wave at the beginning of that element. The constant of proportionality is k. (i) Obtain an expression for the intensity I of the wave at a distance x along the pipe, in terms of the intensity Io at the source end (when x = 0), k and x. [2]

9814 H3 Physics (2026) Oscillations and Wave Motion Tutorial 5 (ii) The intensity of sound at two points 0.80 m apart in a certain pipe used as a speaking -tube is found to be 10 24.8 10 W m−− and 10 24.2 10 W