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Chapter 10: Problem 17

A uniform solid sphere of radius \(R, \operatorname{mass} M,\) and moment of inertia \(I=\frac{2}{5} M R^{2}\) is rolling without slipping along a horizontal surface. Its total kinetic energy is the sum of the energies associated with translation of the center of mass and rotation about the center of mass. Find the fraction of the sphere's total kinetic energy that is attributable to rotation.

Short Answer

Expert verified
The fraction of the sphere's total kinetic energy that is due to rotation is 2/7.

Step by step solution

01

Find the translational kinetic energy

To find the translational kinetic energy, we use the formula: \(T = \frac{1}{2}mv^2\) where \(m\) is the mass of the sphere, and \(v\) is its linear velocity.
02

Find the rotational kinetic energy

To find the rotational kinetic energy, we use the formula: \(R = \frac{1}{2}I\omega^2\) where \(I\) is the moment of inertia of the sphere, and \(\omega\) is its angular velocity.
03

Relate the linear and angular velocities

Since the sphere is rolling without slipping, we can relate the linear velocity \(v\) to its angular velocity \(\omega\): \(v = R\omega\)
04

Express the rotational kinetic energy in terms of linear velocity

Substitute the relation between linear and angular velocities into the expression for rotational kinetic energy: \(R = \frac{1}{2}I(\frac{v}{R})^2 = \frac{1}{2}(\frac{2}{5}mR^2)(\frac{v^2}{R^2}) = \frac{1}{5}mv^2\)
05

Find the total kinetic energy

The total kinetic energy is the sum of translational and rotational kinetic energies: \(K = T + R = \frac{1}{2}mv^2 + \frac{1}{5}mv^2 = \frac{7}{10}mv^2\)
06

Calculate the fraction of rotational kinetic energy in the total kinetic energy

Finally, we find the fraction of the rotational kinetic energy in the total kinetic energy: \(\frac{R}{K} = \frac{\frac{1}{5}mv^2}{\frac{7}{10}mv^2} = \frac{1}{5}\cdot\frac{10}{7} = \boxed{\frac{2}{7}}\) So, the fraction of the sphere's total kinetic energy that is attributable to rotation is \(\frac{2}{7}\).

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Most popular questions from this chapter

Why does a figure skater pull in her arms while increasing her angular velocity in a tight spin?

Consider a cylinder and a hollow cylinder, rotating about an axis going through their centers of mass. If both objects have the same mass and the same radius, which object will have the larger moment of inertia? a) The moment of inertia will be the same for both objects. b) The solid cylinder will have the larger moment of inertia because its mass is uniformly distributed. c) The hollow cylinder will have the larger moment of inertia because its mass is located away from the axis of rotation.

Does a particle traveling in a straight line have an angular momentum? Explain.

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