WJEC Physics for A2: Study and Rev Guide

Angle q in radians (rad) q / rad = arc length radius = s r 2 p rad = 360° Key Term Pointer What’s the point of radians? The degree is an arbitrary unit, because there’s no fundamental reason for choosing 360 degrees in a revolution. By contrast, the radian is a natural unit. Its use simplifies several angle-related formulae. B Express in radians, leaving in the p : 180° , 90° , 45° , 30° . QUICKFIRE QUICKFIRE C A thread 1.32m long is wound round a cylinder. A single turn is 0.48m long. Calculate the angle subtended by the whole thread at the centre of the cylinder. Express it in radians as a multiple of p . QUICKFIRE 3.1 Circular motion We deal with motion in a circle at constant speed, covering ideas of angle measurement and angular velocity, and how to apply F = ma to uniform circular motion. 3.1.1 Defining an angle in radians Imagine an arc of a circle, of any radius, r , centred on O, (where the ‘arms’, OA and OB, of the angle meet). See diagram. Then the angle q is given, in radians, by… q / rad = arc length radius , that is q /rad = s r If s = r , then q /rad = 1 , that is q = 1 radian . Example A ship in the Paci c Ocean cruises for 1200km along the equator. What angle in radians at the centre of the Earth does the journey subtend? (Take the Earth’s equatorial radius to be 6380km .) Answer q = arc length radius = 1200km 6380km = 0.188 rad In practice you’ll probably never have to measure an arc length in order to determine an angle in radians. It’s important, though, to be able to convert between degrees and radians – either way round. Read on. For a complete revolution, s = 2 p r so q / rad = 2 p r r = 2 p . Thus, 2 p rad = 360° , p rad = 180° , 1 rad = 180° p = 57.3° 1° = p 180 rad Example What is 60° in radians? Answer 60° = 60 × 1° = 60 × p 180 rad = p 3 rad = 1.05 rad . The unit ‘rad’ is often omitted when there’s no ambiguity. For example, you may write q = p 3 , meaning q = p 3 rad, but don’t write q = 1.05 . In some of the formulae below, involving angular velocity, ‘rad’ is omitted. s r A B O θ Fig. 3.1.1 q in radians 8 A Level Physics: Study and Revision Guide