Question

# A teenager pushes tangentially on a small hand-drivenmerry-go-round and is able to accelerate it from rest to afrequency of 20 rpm in 8.0s. Assume the

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A teenager pushes tangentially on a small hand-drivenmerry-go-round and is able to accelerate it from rest to afrequency of 20 rpm in 8.0s. Assume the merry-go-round is auniform disk of radius 2.0m and has a mass of 600kg, and twochildren (each with a mass of 20kg) sit opposite each other on theedge.
A) Calculate the torque required to produce theacceleration, neglecting frictional torque.
B) What force is required at the edge?

2021-04-22

The initial angular frequency $$\displaystyle\omega_{{{0}}}={0}$$
The Final angular frequency $$\displaystyle\omega_{{{f}}}={20}{r}\pm$$
$$\displaystyle={20}\cdot{\left({2}\frac{\pi}{{60}}\right)}{r}{a}\frac{{d}}{{s}}$$
The time taken t = 8s
The radius of the merry-go-round R = 2m
Mass of merry-go-round M = 600kg
Mass of child m = 20kg
Therefore the angular acceleration $$\displaystyle\alpha=\triangle\frac{\omega}{\triangle}{t}$$
The Torque required $$\displaystyle{T}={I}_{{{s}{y}{s}{t}{e}{m}}}\cdot\alpha$$
$$\displaystyle={\left({I}_{{{m}{e}{r}{r}{y}}}+{I}_{{chi{l}{d}{r}{e}{n}}}\right)}\cdot\alpha$$
$$\displaystyle={\left[{\left(\frac{{1}}{{2}}\right)}{M}_{{{m}{e}{r}{r}{y}}}{R}^{{{2}}}+{2}{m}_{{chi{l}{d}}}{R}^{{{2}}}\right]}\cdot\alpha$$
substitute the values and calculate the Torque.
b) The Force required at the edge is calculated from the torque
$$\displaystyle{T}={F}_{{{p}{e}{r}{p}{e}{n}{d}{i}{c}\underline{{a}}{r}}}\cdot{R}{a}{d}{i}{u}{s}$$
Or $$F=\frac{t}{R}$$