Circular Motion Radians An angle in radians is defined as the ratio of the arc length to the radius. s r r  (radians) =arc length / radius  = s / r  = s / r s = r  Note Remember that  radians= 180 0

Consider a particle moving around a circular path from A to B as shown.  O A B Angular displacement and Angular velocity Angular displacement,  is the angle swept out by the radius vector as it moves from A to B.  is measured in radians. Angular velocity is defined as the rate of change of angular displacement.  = d  / dt (in rads -1 ) For constant angular velocity:  =  / t  =  / t

The period of rotation, T is the time for one complete revolution.  =  / t = 2  / T = 2  / T T = 2  /  Frequency, f is the number of rotations per second: f = 1 / T (can be rearranged to T = 1 / f) Then substituting : 1 / f = 2  /  f =  / 2   = 2  f

Example A rotor is spinning at 3000r.p.m. Find: (a)The angular velocity. (b)The period of rotation. Solution: (a)  =  / t  = (3000 x 2  ) / 60 = 100  rads -1 = 100  rads -1 (b)T = 2  /  = 2  / 100  = 2  / 100  = 0.02s = 0.02s

Uniform motion in a Circle Consider a particle moving with uniform speed in a circular path as shown below.  s r v  The rotational speed, v is constant as is the angular velocity, . T is the period of the motion and is the time taken to cover 2  radians.  = 2  / T But, v = 2  r / T => v = r  NOTE: The rotational speed is also know as the tangential speed, v. It is always perpendicular to the radius of the circle

Angular Acceleration In this case the radius vector sweeps out increasing angular displacements in successive time intervals. 11 22 33 44 The angular acceleration, α is defined as: The rate of change of angular velocity. α = d  / dt = d 2 θ / dt 2 α = d  / dt = d 2 θ / dt 2 We can use calculus methods to obtain equations for angular motion with constant acceleration.

ω = ω 0 + αt ω 2 = ω αθ θ = ω 0 t + ½ αt 2

Example A wheel rotates through 12 revolutions in 4s. (a) Find its angular displacement (b) Calculate the angular velocity (c) Find the angular deceleration if it is now brought to rest in 3s. (a)θ = 12 X 2  = 24  rad = 24  rad (b)θ= 24  rad (b)θ = 24  rad  = ? t = 4s  = θ  = θ / t 24  = 24  / 4  rads -1 = 6  rads -1

ω 0 = 6  rads -1 ω = 0 rads -1 ω = 0 rads -1 α = ? α = ? t = 3s t = 3s ω = ω 0 + αt ω = ω 0 + αt 0 = 6  + 3 α 0 = 6  + 3 α α = -6  / 3 α = -6  / 3 = -2  rads -2 = -2  rads -2

Tangential acceleration If an object is moving with angular acceleration, then its tangential velocity, v is changing. The tangential acceleration, a t is given by: a t = dv / dt But, v = r ω=> = drω / dt = r dω / dt = r dω / dt a = r α