WJEC Physics for A2: Study and Rev Guide

Simple harmonic motion occurs when an object moves so that its acceleration is always directed towards a fixed point and is proportional to its displacement from that point. Mathematically: a = − w 2 x The spring constant , k is the force exerted per unit extension by a spring. UNIT: N m −1 Key Terms Pointer When the block in Fig. 3.2.1 is to the left of the equilibrium position, the spring is compressed. As long as the turns of the spring don’t touch, k will have the same value for compression as extension. B Calculate the gradient of a graph of a against x (see Fig. 3.2.2) for a body of mass 0.15kg oscillating on a spring for which k = 5.4Nm −1 . QUICKFIR QUICKFIRE C Show that the units of w are s −1 . What quantity associated with oscillations has the same unit? QUICKFIR 3.2 Vibrations When an object is moving to and fro regularly about a xed point, we say that it is vibrating or oscillating . We’ll deal mainly with two well-known examples: an object on a spring, and a pendulum. Both these systems can oscillate naturally with so-called simple harmonic motion (shm). 3.2.1 The definition of simple harmonic motion It helps to have a particular system in mind. We’ll consider the block on a spring, shown in Fig. 3.2.1. When the block, m , is displaced from its equilibrium position and released, it oscillates back and forth. EQUILIBRIUM block end of spring fixed low friction guide rails m x DISPLACED m Fig. 3.2.1 Block oscillating on spring When m ’s displacement is x , the spring’s extension is x , and, according to Hooke’s law, the spring exerts a force proportional to x on the block, given by: F = − kx . ■ The minus sign is because the force is in the opposite direction to x . ■ k is the spring constant : see Key terms . Assuming resistive forces are negligible, the block’s acceleration is: a = F m so a = − k m x which is often written as a = − w 2 x . in which w is a constant given by w = k m . Oscillations obeying a = − w 2 x , that is a = − constant × x , are called simple harmonic motion (shm). This de nition is put into words in Key terms . The relationship is shown as a graph in Fig. 3.2.2. A is the amplitude : the maximum value of the displacement. a ω 2 A A – A x – ω 2 A Fig. 3.2.2 Graph of a = − w 2 x 13 3.2 Vibrations