Chapter 17: Circular motion Chapter outline ■ ■ ■ ■
use degrees and radians as measures of angle, including the expression of angular displacement in radians explain uniform circular motion in terms of a centripetal force causing a centripetal acceleration solve problems involving angular displacement and velocity, including use of the equation v = rω 2 solve problems involving centripetal force and acceleration, including use of the equations a = rω r 2 = v and r 2 F mrω mr 2 = mv r
KEY TERMS
radian: a unit of angle such that 2π radians = 360° angular displacement θ: the angle through which an object moves in a circle angular velocity ω: the rate at which the angular displacement changes centripetal force: the resultant force acting on an object that is moving in a circle Equations: angular velocity =
Δθ angular displacement ; ω= Δt time
speed = radius × angular velocity; v = r ω 106
2 2 centripetal acceleration = radius × (angular velocity ) ; a = r ω 2 = ν r
centripetal force = mass × centripetal acceleration; F = mr ω 2 =
mν 2 r
Exercise 17.1 Angular measure The radian is a more ‘natural’ unit of measurement of angles than the degree. Angles in radians can be calculated knowing the length s of the arc subtended by the angle and the radius r of the circle: θ = rs . This exercise provides practice in calculating angles in radians and converting between degrees and radians. 1 For each diagram a–f, calculate the unknown quantity θ, s or r, from the other two: s 1.50 m θ
2.0 rad 0.5
1.00 m
a
b
1.0 m
cm
0.50 r
c