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Ch 12: Rotation of a Rigid Body
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
All textbooksKnight Calc 5th EditionCh 12: Rotation of a Rigid BodyProblem 36
Chapter 12, Problem 36

A solid sphere of radius R is placed at a height of 30 cm on a 15° slope. It is released and rolls, without slipping, to the bottom. From what height should a circular hoop of radius R be released on the same slope in order to equal the sphere's speed at the bottom?

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Step 1: Begin by analyzing the energy conservation principle for both the solid sphere and the circular hoop. The total mechanical energy at the top of the slope is purely potential energy, and at the bottom, it is converted into both translational kinetic energy and rotational kinetic energy.
Step 2: Write the energy conservation equation for the solid sphere. The potential energy at the top is \( mgh \), and the kinetic energy at the bottom is \( \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2 \), where \( I \) is the moment of inertia of the sphere and \( \omega \) is the angular velocity. For a solid sphere, \( I = \frac{2}{5}mR^2 \) and \( \omega = \frac{v}{R} \). Substitute these values into the equation.
Step 3: Write the energy conservation equation for the circular hoop. The potential energy at the top is \( mgh \), and the kinetic energy at the bottom is \( \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2 \). For a hoop, \( I = mR^2 \) and \( \omega = \frac{v}{R} \). Substitute these values into the equation.
Step 4: Compare the final speeds of the sphere and the hoop at the bottom of the slope. The speed of the sphere is determined by its energy conservation equation, and the speed of the hoop is determined by its own energy conservation equation. Set the speeds equal to each other to find the height \( h \) from which the hoop must be released.
Step 5: Solve the resulting equation for \( h \) in terms of the given height of the sphere (30 cm). This involves algebraic manipulation to isolate \( h \) for the hoop, ensuring that the speeds match at the bottom of the slope.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Conservation of Energy

The principle of conservation of energy states that the total energy in a closed system remains constant. In this scenario, the potential energy of the objects at the top of the slope is converted into kinetic energy as they roll down. For both the solid sphere and the circular hoop, the initial gravitational potential energy must equal the sum of translational and rotational kinetic energy at the bottom.
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Moment of Inertia

Moment of inertia is a measure of an object's resistance to changes in its rotation. Different shapes have different moments of inertia, which affects how they roll down a slope. For a solid sphere, the moment of inertia is (2/5)MR², while for a circular hoop, it is MR². This difference influences the distribution of energy between translational and rotational motion.
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Rolling Without Slipping

Rolling without slipping occurs when an object rolls on a surface without sliding, meaning the point of contact with the surface is momentarily at rest. This condition relates the linear speed of the center of mass to the angular speed of the object. For both the sphere and the hoop, this relationship is crucial for determining their speeds at the bottom of the slope, as it affects how energy is distributed between translational and rotational forms.
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