TJC 7 Circular Motion

Student’s Copy Unit 6: MO Unit 7: Circular Motion Learning Outcomes Students should be able to: 7.1 Kinematics of uniform circular motion 7(a) express angular displacement in radians 7(b) show an understanding of and use the concept of angular velocity 7(c) recall and use v = r to solve problems 7.2 Centripetal acceleration 7(d) show an understanding of centripetal acceleration in the case of uniform motion in a circle, and qualitatively describe motion in a curved path (arc) as due to a resultant force that is both perpendicular to the motion and centripetal in direction 7(e) recall and use centripetal acceleration a = r2 and a = 2v r to solve problems 7(f) recall and use F = mr2 and F = 2mv r to solve problems.

2025 Temasek Junior College 2 1 Introduction We can find many examples of circular motion around us, e.g. bicycles and cars going round a corner and the Moon orbiting around the Earth. In the first part of this topic, we will describe the motion of objects moving in a circular path; in the subsequent parts, we will consider what causes motion to be circular. 2 Definition of Physical Quantities 2.1 Angular Displacement,  LO(a) In terms of the length of arc s and the radius of the circle r, angular displacement θ is expressed as       = s r  Angular displacement is measured in radian (rad).  For one complete revolution, θ = 2r r = 2 rad (i.e. 2 rad = 360o)  To convert θ in degrees to radians: θ(rad) 2π = θ(°) 360° Note: The radian is physically dimensionless as it is the ratio of two lengths. 2.2 Angular Velocity,  LO(b),(c) i.e.  = dθ dt unit of ω: rad s-1 Consider an object moving with constant speed v in a circular path of radius r. In time t the object moves along an arc of length s and sweeps out an angle .   = ∆θ ∆t   = ∆s r⁄ ∆t (since s = r)   = 1 r ∆s ∆t  = v r (since v = ∆s ∆t) Hence v = r One radian is defined as the angle subtended at the centre of a circle by an arc equal in length to the radius of the circle. Angular velocity is defined as the rate of change of angular displacement s  r

2025 Temasek Junior College 3 A B C 12 6 9 3 For uniform circular motion, since v has the same magnitude throughout the motion, angular velocity is a constant (equal angle is swept out in equal time intervals). If the time taken to complete one revolution (i.e. to turn through an angle of 2 radians) is T, then T is known as the period. Note: Angular velocity is a vector quantity. Example 1 (a) Which point A, B or C on the minute hand of a clock is moving with the greatest speed? (b) Calculate the angular velocity of the minute hand. Example 2 A model car moves round a circular track of radius 0.30 m at 2.0 revolutions per second. Calculate