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### Fundamentals Of Physics

Book edition 10th Edition
Author(s) David Halliday
Pages 1328 pages
ISBN 9781118230718

# In a playground, there is a small merry-go-round of radius 1.20 m and mass 180 kg. Its radius of gyration (see Problem 79 of Chapter 10) is 91.0 cm. A child of mass 44.0 kg runs at a speed of 3.00 m/s along a path that is tangent to the rim of the initially stationary merry-go-round and then jumps on. Neglect friction between the bearings and the shaft of the merry-go-round. Calculate (a) the rotational inertia of the merry-go-round about its axis of rotation, (b) the magnitude of the angular momentum of the running child about the axis of rotation of the merry-go-round, and (c) the angular speed of the merry-go-round and child after the child has jumped onto the merry-go-round.

1. Rotational Inertia of merry go round about its axis of rotation is 149 kg.m2.
2. The magnitude of the angular momentum of the running child about the axis of rotation of the merry-go-round is 158 kg.m2/s.
3. The angular speed of the merry-go-round and child after the child has jumped onto the merry-go-round is 0.744 rad/s2.
See the step by step solution

## Step 1: Given

$\mathrm{m}=180\mathrm{kg}\phantom{\rule{0ex}{0ex}}\mathrm{k}=91\mathrm{cm}\phantom{\rule{0ex}{0ex}}\mathrm{M}=44\mathrm{kg}\phantom{\rule{0ex}{0ex}}\mathrm{r}=1.20\mathrm{m}\phantom{\rule{0ex}{0ex}}\mathrm{v}=3\mathrm{m}/\mathrm{s}$

## Step 2: Determining the concept

Use the formula for rotational inertia in terms of mass and radius of gyration to find the rotational inertia. Using the formula for angular momentum in terms of mass, velocity and radius, find the angular momentum. Finally, use conservation of angular momentum to find angular velocity. According to the conservation of momentum, momentum of a system is constant if no external forces are acting on the system.

Formula are as follow:

${\mathbf{I}}{\mathbf{=}}{\mathbf{m}}{\mathbf{×}}{{\mathbf{k}}}^{{\mathbf{2}}}\phantom{\rule{0ex}{0ex}}{\mathbf{L}}{\mathbf{=}}{\mathbf{m}}{\mathbf{×}}{\mathbf{v}}{\mathbf{×}}{\mathbf{r}}$

where,r is radius, v is velocity, m is mass, L is angular momentum, l is moment of inertia and r is radius of gyration.

## Step 3: Determining the rotational Inertia of merry go round about its axis of rotation

(a)

To find rotational inertia, use the following formula:

$\mathrm{I}=\mathrm{m}×{\mathrm{k}}^{2}\phantom{\rule{0ex}{0ex}}\mathrm{I}=180×0.{91}^{2}\phantom{\rule{0ex}{0ex}}\mathrm{I}=149\mathrm{kg}.{\mathrm{m}}^{2}$

Hence, rotational Inertia of merry go round about its axis of rotation is 149 kg.m2.

## Step 4:  Determining the magnitude of the angular momentum of the running child about the axis of rotation of the merry-go-round

(b)

Now, to find angular momentum,

$\mathrm{L}=\mathrm{M}×\mathrm{r}×\mathrm{v}\phantom{\rule{0ex}{0ex}}\mathrm{L}=44×3×1.2$

So,

data-custom-editor="chemistry" $\mathrm{L}=158\mathrm{kg}.{\mathrm{m}}^{2}/\mathrm{s}$

Hence, the magnitude of the angular momentum of the running child about the axis of rotation of the merry-go-round is 158 kg.m2/s.

## Step 5:  Determining the angular speed of the merry-go-round and child after the child has jumped onto the merry-go-round

(c)

Now, to find angular velocity, use conservation of momentum,

${\mathrm{L}}_{\mathrm{f}}={\mathrm{L}}_{\mathrm{child}}\phantom{\rule{0ex}{0ex}}{\mathrm{L}}_{\mathrm{f}}=\left(\mathrm{I}+{\mathrm{mr}}^{2}\right)\mathrm{\omega }\phantom{\rule{0ex}{0ex}}{\mathrm{L}}_{\mathrm{child}}=\mathrm{mvr}\phantom{\rule{0ex}{0ex}}\mathrm{mvr}=\left(\mathrm{I}+{\mathrm{mr}}^{2}\right)\mathrm{\omega }\phantom{\rule{0ex}{0ex}}\mathrm{\omega }=\frac{\mathrm{mvr}}{\mathrm{I}+{\mathrm{mr}}^{2}}$

So,

$\mathrm{\omega }=\frac{158}{149+44+1.{2}^{2}}=0.744\mathrm{rad}/\mathrm{s}\phantom{\rule{0ex}{0ex}}\mathrm{\omega }=0.744\mathrm{rad}/\mathrm{s}$

Hence, the angular speed of the merry-go-round and child after the child has jumped onto the merry-go-round is 0.744 rad/s2.

Therefore, use the formula for rotational inertia in terms of mass and radius of gyration to find the rotational inertia. Using the formula for angular momentum in terms of mass, velocity and radius, the angular momentum can be found. Finally, conservation of angular momentum can be used to find angular velocity.

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