Question

Two children of the same mass m = 22:4 kg are playing on the rotating carousel below. The oor of
the carousel has mass M = 35:0 kg and radius R = 1:75 m. The centre of mass of each child is at a
distance of r = 30:5 cm from the centre of the carousel. The carousel is rotating at an angular velocity
of ! = 1:05 rad/s.
(a) (1 point) What is the tangential speed of the children?
(b) (1 point) What is the magnitude of the centripetal force on each child?
(c) (2 points) What is the angular momentum of the whole rotating system (carousel + children)?
(Ignore the mass of the bars and treat the children as point masses.)
(d) (3 points) A moment later, 2 more children of the same mass as the original two jump onto the
edge of the carousel on opposite sides. What is the new angular velocity of the whole system?
(Assume the new children are also point masses.)
(e) (1 points) After time t = 10s, the whole system (carousel + 4 kids) slows down to a stop. What
torque did friction exert on the system?

Answer 1

(a) Tangential speed of children v = = 1.05 30.5 10-2 = 0.32025 m/s

(b) magnitude of centripetal force = ( m v2 / r ) = 22.4 0.3205 0.3205 / ( 30.5 10-2 )

                                                      = 7.544 m/s2

(c) Angular momentum = I + 2 m r2

where I is moment of inertia of carousel , is angular speed of rotation of carousel,

m is mass of child and r is distance between centre of carousel

By considering the carousel as a circular disc of mass M = 35 kg and radius R = 1.75 m,

Moment of inertia I = (1/2) MR2 = (1/2) 35 1.75 1.75 53.6 kg-m2

Hence total angular momentum = 53.6 + ( 2 22.4 1.75 0.305 0.305) 60.9 kg-m2

(d) new angular speed 1 of rotation is determined by conservation of angular momentum

( I + 2 m r2 ) o = ( I + 4 m r2 ) 1

1 = 60.9 / [ 53.6 + ( 4 22.4 0.305 0.305 ) ] = 0.983 rad /s

(e) If whole system comes to rest after 10 s, retardation of angular motion = / t

retardation = 0.983 / 10 = 0.0983 rad / s2

Torque = I = ( IC + 4mr2 ) = ( 53.6 + 4 22.4 0.305 0.305 ) 0.0983 6.1 N-m

where moment of inertia I is sum of moment of inertia of carousel and moment of inertia due to 4 children .