Question

A body is initially (at t=0s) rotating about a z-axis as shown below. Its initial angular speed is 8.50 rad/s. It's rotation is slowing at a constant rate of 0.150 rad/s^2. a) How long does it take to stop? b) How many revolutions did it make in this time period? c) If the point shown is 17.0 cm from the z-axis, what was the initial speed at this point and what was the initial acceleration magnitude at this point?

Answer 1

here,

initial angular speed , w0 = 8.5 rad/s

the angular accelration , alpha = - 0.15 rad/s^2

a)

let the time taken to stop be t

w = w0 + alpha * t

0 = 8.5 - 0.15 * t

solving for t

t = 56.67 s

b)

the angle covered ,theta = w0^2 /( 2*alpha)

theta = 8.5^2 /( 2 * 0.15) = 240.8 degree

the number of revolutions , N = theta /2pi

N = 38.3 rev

c)

r = 17 cm = 0.17 m

the initial speed , u = r * w0

u = 0.17 * 8.5 = 1.45 m/s

the tangential accelration , at = r * alpha = 0.0255 m/s^2

the initial centripital accelration , ac = w0^2 * r = 12.2825 m/s^2

the magnitude of initial accelration , a = sqrt(at^2 + ac^2) = 12.3 m/s^2