PRACTICE PROBLEM $-$ PREDICT$/$CALCULATE

When the ax is pressed firmly against the grindstone for sharpening, the angular speed of the grindstone decreases.

$($a$)$ If the rotational kinetic energy of the grindstone is cut in half to $6\mathrm{.}50$ J $,$ is the corresponding angular speed greater than, less than, or equal to one$-$half the original angular speed of $2\mathrm{.}46ra\frac{d}{s}?$ Explain. $($b$)$ Find the new angular speed. $[$Answer: $($a$)$ The rotational kinetic energy depends on the square of the angular speed. Therefore, reducing the kinetic energy by a factor of $2$ corresponds to reducing the angular speed by a factor of $\sqrt[\phantom{2}]{2}.$ Thus, the new angular speed is greater than $\frac{1}{2}$ the initial angular speed. $($b$)$ The new angular speed $($using $K=6\mathrm{.}50J)$ is $=\sqrt[\phantom{2}]{2\frac{K}{\text{I}}}=1\mathrm{.}74ra\frac{d}{s},$ which is greater than $\frac{1}{2}(2\mathrm{.}46ra\frac{d}{s})=1\mathrm{.}23ra\frac{d}{s}.]$

Some related homework problems: Problem $47,$ Problem $48,$ Problem $51$