Question

a 50.0 kg child is on a 200. kg merry-go-round with a radius of 1.50 m. the child starts at 0.5 m from the center and moves to the edge. the child's parent accelerates the merry-go-round from rest at 0.75 rad/s^2 to a speed of 1.5 rad/s. How long did the parent push the merry-go-round? how much force is required to hold the child when at 0.5 m from the center? determine the kinetic energy of the system when moving at 1.5 rad/s. determine the final angular velocity when the child is at the edge.

Answer 1

here,

mass of child , m1 = 50 kg

mass of merry-go-round , m2 = 200 kg

radius , R = 1.5 m

r1 = 0.5 m

angular accelration , alpha = 0.75 rad/s^2

final angular speed , w = 1.5 rad/s

a)

let the time taken to accelrate be t

w = 0 + alpha * t

1.5 = 0 + 0.75 * t

t = 2 s

the time taken is 2 s

b)

the force required to hold the child , F1 = m * w^2 * r1

F1= 50 * 1.5^2 * 0.5 N

F1 = 56.25 N

c)

the kinetic energy of the system , KE = 0.5 * (m1 * r1^2 + 0.5 * m2 * R^2 ) * w^2

KE = 0.5 * (50 * 0.5^2 + 0.5 * 200 * 1.5^2) * 1.5^2

KE = 267.2 J

d)

let the new angular speed be w'

using conservation of angular momentum

(m1 * r1^2 + 0.5 * m2 * R^2 ) * w= (m1 * R^2 + 0.5 * m2 * R^2 ) * w'

(50 * 0.5^2 + 0.5 * 200 * 1.5^2) * 1.5 = (50 * 1.5^2 + 0.5 * 200 * 1.5^2) * w'

solving for w'

w' = 1.06 rad/s

the new angular speed is 1.06 rad/s