Question

A 1.60 m radius playground merry-go-round has a mass of 120 kg and is rotating with an angular velocity of 0.58 rev/s. what is its angular velocity, in rev/s, after a 18kg child gets onto it by grabbing its outer edge? the child is initially at rest.

Answer 1

We are given the following information

Radius of merry-go-round = 1.60 m

Mass of merry-go-round = 120 kg

Angular velocity of merry-go-round = 0.58 rev/s

Mass of child = 18 kg

We are asked to find the angular velocity of the merry-go-round after the child gets onto it by grabbing its outer edge.

The angular momentum before and after the child gets onto the merry-go-round must be equal

`L_b=L_a`

The angular momentum before the child gets onto the merry-go-round is the product of the moment of inertia of the merry-go-round without the child and the angular velocity of the merry-go-round before the child gets onto it.

The angular momentum after the child gets onto the merry-go-round is the product of the moment of inertia of the merry-go-round with the child and the final angular velocity of the merry-go-round after the child gets onto it.

`I_m\cdot\omega_b=I_(m+c)\cdot\omega_f`

Arranging the equation for the final angular velocity of the merry-go-round,

`\begin{gathered} \omega_f=(I_m\cdot\omega_b)/(I_(m+c)) \\ \omega_f=((m_mr^2_m)/(2)\cdot\omega_b)/((m_mr^2_m)/(2)+m_cr^2_c) \end{gathered}`

Where

mm = Mass of merry-go-round = 120 kg

rm = Radius of merry-go-round = 1.60 m

mc = Mass of child = 18 kg

rc = Radius of child relative to merry-go-round = 1.60 m

wb = Angular velocity of merry-go-round = 0.58 rev/s

First, convert the angular velocity from rev/s to rad/s

`0.58*2\pi=3.64\; \; \frac{\text{rad}}{s}`

So, the final angular velocity is

`\begin{gathered} \omega_f=((m_mr^2_m)/(2)\cdot\omega_b)/((m_mr^2_m)/(2)+m_cr^2_c) \\ \omega_f=\frac{(120\cdot1.60^2)/(2)\cdot3.64}{(120\cdot1.60^2)/(2)+18\cdot_{}1.60^2_{}} \\ \omega_f=2.80\; \; (rad)/(s) \end{gathered}`

Therefore, the angular velocity of the merry-go-round after the child gets onto it by grabbing its outer edge is 2.80 rad/s