Physics
Physics
5th Edition
ISBN: 9781260486919
Author: GIAMBATTISTA
Publisher: MCG
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Chapter 8, Problem 84P
To determine

The speed of merry goes round after child gets into it. How much the rotational kinetic energy change.

Expert Solution & Answer
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Answer to Problem 84P

The speed of merry goes round after child gets into it is ωf=0.61rev/s_, and change in kinetic energy is ΔKr=660J_.

Explanation of Solution

Write the expression for initial rotational inertia before child enters to the merry go round

Ii=12MR2 (I)

Here, M is the mass of merry go round, R is the radius of the merry go round

After the child enters into merry go round, the rotational inertia changes. It is required to take account of the weight of the child.

Write the expression for the rotational inertia after the child enters the merry go round

If=12MR2+FgR2g (II)

Here, Fg is the weight of the child, g is the acceleration due to gravity

Write the expression of initial angular momentum

Li=Iiωi (III)

Here, Li is the initial angular momentum, ωi is the initial angular speed

Substitute equation (I) in (III)

Li=12MR2ωi (IV)

Write the expression of final angular momentum

Lf=Ifωf (V)

Here, Lf is the final angular momentum, ωf is the final angular speed

Substitute equation (II) in (V)

Lf=(12MR2+FgR2g)ωf (VI)

According to law of conservation of angular momentum, Li, and Lf are same. Equate equation (VI) and (IV)

Li=Lf12MR2ωi=(12MR2+FgR2g)ωf (VII)

Rearrange equation (VII) to obtain an expression for ωf

12MR2ωi=(12MR2+FgR2g)ωfωf=12MR2ωi12MR2+FgR2g=(1+2FggM)1ωi (VIII)

Write the expression for change rotational kinetic energy

ΔKr=12Ifωf212Iiωi2 (IX)

Substitute equation (I) and (II) in (IX)

ΔKr=12(12MR2+FgR2g)ωf212(12MR2)ωi2 (X)

Conclusion:

Substitute 180N for Fg, 9.8m/s2 for g, 160kg for M, and 0.75rev/s ωi in equation (VIII)

ωf=(1+2(180N)9.8m/s2×160kg)10.75rev/s=0.6099rev/s0.61rev/s

Substitute 180N for Fg, 9.8m/s2 for g, 160kg for M, 2.0m for R, 0.75rev/s ωi, and 0.61rev/s for ωf in equation (X)

ΔKr=12{(12(160kg)(2.0m)2+(180N)(2.0m)29.8m/s2)(0.61rev/s)212(12(160kg)(2.0m)2)(0.75rev/s)2}(2πradrev)2=662J

Therefore, the speed of merry goes round after child gets into it is ωf=0.61rev/s_, and change in kinetic energy is ΔKr=660J_.

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