Question 17.SP.3: A sphere, a cylinder, and a hoop, each having the same mass ......
A sphere, a cylinder, and a hoop, each having the same mass and the same radius, are released from rest on an incline. Determine the velocity of each body after it has rolled through a distance corresponding to a change in elevation h.
The problem will first be solved in general terms, and then results for each body will be found. We denote the mass by m, the centroidal moment of inertia by \bar{I}, the weight by W, and the radius by r.
Kinematics. Since each body rolls, the instantaneous center of rotation is located at C and we write
\omega=\frac{\bar{v}}{r}
Kinetic Energy
\begin{aligned}T_1 & =0 \\T_2 & =\frac{1}{2} m \bar{v}^2+\frac{1}{2} \bar{I} \omega^2 \\& =\frac{1}{2} m \bar{v}^2+\frac{1}{2} \bar{I}\left(\frac{\bar{v}}{r}\right)^2=\frac{1}{2}\left(m+\frac{\bar{I}}{r^2}\right) \bar{v}^2\end{aligned}
Work. Since the friction force F in rolling motion does no work,
U_{1 \rightarrow 2}=W h
Principle of Work and Energy
\begin{aligned}T_1+U_{1 \rightarrow 2} & =T_2 \\0+W h & =\frac{1}{2}\left(m+\frac{\bar{I}}{r^2}\right) \bar{v}^2 \quad \bar{v}^2=\frac{2 W h}{m+\bar{I} / r^2}\end{aligned}
Noting that W = mg, we rearrange the result and obtain
\bar{v}^2=\frac{2 g h}{1+\bar{I} / m r^2}
Velocities of Sphere, Cylinder, and Hoop. Introducing successively the
particular expression for \bar{I}, we obtain
Sphere: \bar{I}=\frac{2}{5} m r^2 \quad \bar{v}=0.845 \sqrt{2 g h}
Cylinder: \bar{I}=\frac{1}{2} m r^2 \quad \bar{v}=0.816 \sqrt{2 g h}
Hoop: \bar{I}=m r^2 \quad \bar{v}=0.707 \sqrt{2 g h}
Remark. Let us compare the results with the velocity attained by a frictionless block sliding through the same distance. The solution is identical to the above solution except that ω = 0; we find \bar{v}=\sqrt{2 g h}.
Comparing the results, we note that the velocity of the body is independent of both its mass and radius. However, the velocity does depend upon the quotient \bar{I} / m r^2=\bar{k}^2 / r^2, which measures the ratio of the rotational kinetic energy to the translational kinetic energy. Thus the hoop, which has the largest \bar{k} for a given radius r, attains the smallest velocity, while the sliding block, which does not rotate, attains the largest velocity.
