Motion in a Circle JPJC Notes

JURONG PIONEER JUNIOR COLLEGE 9749 H2 PHYSICS MOTION IN A CIRCLE Content (I) Kinematics of uniform circular motion (II) Centripetal acceleration (III) Centripetal force Learning Outcomes Candidates should be able to: (a) express angular displacement in radians. (b) show an understanding of and use the concept of angular velocity to solve problems. (c) recall and use vr = to solve problems. (d) describe qualitatively motion in a curved path due to a perpendicular force, and understand the centripetal acceleration in the case of uniform motion in a circle. (e) recall and use centripetal acceleration 2ar = , and 2va r= to solve problems. (f) recall and use centripetal force 2F mr = , and 2mvF r= to solve problems.

2 Introduction ▪ In the previous topic on Kinematics, we learnt about uniformly accelerated motion. An object accelerating along a straight line experiences a net force that acts along its direction of motion. However, if a constant net force acts at an angle to the direction of motion at any instant, the object moves in a curved path (e.g. projectile motion). ▪ In this topic, we will study the circular motion of objects in which the acceleration is not uniform. In particular, we will focus on uniform circular motion in which an object travels at a constant speed. 1 Kinematics of uniform circular motion (a) Candidates should be able to express angular displacement in radians. 1.1 Angular displacement ▪ Consider an object movin g from A to B in a circle with uniform speed v round a fixed point O as centre. Fig. 1 Object moving in uniform circular motion from A to B ▪ The angle  swept through by the radius is known as the angular displacement. ▪ It is defined by the equation s r = where s is the arc length AB and r is the radius of the circle. ▪ The angular displacement  is measured in radians. One radian (rad) is defined as the angle subtended at the centre of a circle by an arc equal in length to the radius. r v v O A B r

3 (b) Candidates should be able to show an understanding of and use the concept of angular velocity to solve problems. 1.2 Angular velocity ▪ Angular velocity  is the rate of change of angular displacement. d dt  = It is measured in rad s−1. ▪ For an object moving with constant angular velocity, t  = . Example 1 A boy seated 2.0 m away from the centre of a merry -go-round completes one -sixth of a revolution in 3.0 s. Calculate (a) his angular displacement  from his starting position, (b) the distance he travels from his starting position, (c) his angular velocity. Solution: (a) For one revolution,  = 2 rad; for one-sixth of a revolution, angular displacement  1 26 = = 1.047 1.0= rad (b) The distance travelled is the arc length s of the circle. s r= 2.0 1.047= 2.1= m (