Attempt $1$

An empty cylindrical barrel is open at one end and rolls without slipping straight down a hill. The barrel has a mass of $23\mathrm{.}0$ kg $,$ a radius of $0\mathrm{.}400$ m $,$ and a length of $0\mathrm{.}600$ m $.$ The mass of the end of the barrel equals a third of the mass of its side, and the thickness of the barrel is negligible.

The acceleration due to gravity is $g=9\mathrm{.}80\frac{m}{s^2}.$

What is the translational speed $v_f$ of the barrel at the bottom of the hill if released from rest $19\mathrm{.}0$ m above the bottom?

$v_f={}_{\text{I}\text{n}\text{c}\text{o}\text{r}\text{r}\text{e}\text{c}\text{t}\text{A}\text{n}\text{s}\text{w}\text{e}\text{r}}\frac{m}{s}$

Suppose a lid was added to the barrel that had the same mass as its other end. Would adding the lid to the barrel increase, decrease, or not change the barrel's speed at the bottom of the hill if the barrel was released from the same height? Explain.

The speed would decrease because the barrel's overall moment of inertia is increased, resulting in more of the initial potential energy being converted to rotational rather than translational kinetic energy.

The speed would increase because the distribution of the additional mass results in a fractional increase in the barrel's translational kinetic energy over its rotational kinetic energy.