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Rank the order of the collection of objects.
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Answer to Problem 98P
The objects reach the bottom in the following order from cube, solid sphere, solid cylinder, hollow sphere,and hollow cylinder.
Explanation of Solution
Write the equation of conservation of energy.
Here,
Write the equation for initial potential energy of the object.
Here,
Write the equation for final translational kinetic energy of the object.
Here,
Write the equation for final rotational kinetic energy of the object.
Here,
Write the equation for moment of inertia of the solid sphere.
Here,
Write the relatin between linear velocity and angular velocity.
Write the equation for moment of inertia of the hollow sphere.
Write the equation for moment of inertia of the solid cylinder.
Write the equation for moment of inertia of the hollow cylinder assumed as a thin wall like a hoop.
Conclusion:
Solid sphere:
Substitute equation (II), (III), (IV), (V) and (VI) in equation (I).
Rewrite the above equation for final speed of the object.
Hollow sphere:
Substitute equation (II), (III), (IV), (VII) and (VI) in equation (I).
Rewrite the above equation for final speed of the object.
Solid cylinder:
Substitute equation (II), (III), (IV), (VIII) and (VI) in equation (I).
Rewrite the above equation for final speed of the object.
Hollow cylinder:
Substitute equation (II), (III), (IV), (IX) and (VI) in equation (I).
Rewrite the above equation for final speed of the object.
Cube:
Substitute equation (III) and (II) in equation (I).
Since
Therefore, the objects reach the bottom in the following order from cube, solid sphere, solid cylinder, hollow sphere,and hollow cylinder.
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