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Chapter 10: Problem 6

A solid sphere and a hollow sphere of the same mass and radius are rolling along level ground. If they have the same total kinetic energy, which is moving faster?

Short Answer

Expert verified
By solving the equation, we find that \( v_h = \sqrt{\frac{7}{5}} v_s > v_s \) which means the hollow ball is moving faster than the solid ball.

Step by step solution

01

Writing down the kinetic energy formulas for both spheres

Because the two spheres have the same kinetic energy, we can say \( \frac{1}{2}m_s v_s^2 + \frac{1}{2}I_s \omega_s^2 = \frac{1}{2}m_h v_h^2 + \frac{1}{2}I_h \omega_h^2 \) where the subscripts s and h refer to solid and hollow, respectively. Because the spheres are rolling without slipping, we can say \( v_s = r\omega_s \) and \( v_h = r\omega_h \). So \( I_s = \frac{2}{5}mr^2 \) and \( I_h = \frac{2}{3}mr^2 \).
02

Substituting the values of moment of inertia in the formulas

Substitute \( I_s = \frac{2}{5}mr^2 \) and \( I_h = \frac{2}{3}mr^2 \) into the kinetic energy equations to get \( \frac{1}{2}m_s v_s^2 + \frac{1}{5}m_s v_s^2 = \frac{1}{2}m_h v_h^2 + \frac{1}{3}m_h v_h^2 \).
03

Simplify the equation

Simplify the equation to find the relationship between the velocities of the two spheres. This leads to \( \frac{7}{10}m_s v_s^2 = \frac{5}{6}m_h v_h^2 \). Because both the solid sphere and the hollow sphere have the same mass, \( m_s = m_h \), we can cancel out the mass in the equation. This gives \( \frac{7}{10} v_s^2 = \frac{5}{6} v_h^2 \). Solve this equation for \( v_h \) in terms of \( v_s \) to determine which one is moving faster.

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