Question

A wheel is rotaing freely with an angular speed $\omega$ on a shaft. The moment of inertia of the wheel is $I$ and the moment of inertia of the shaft is negligible. Another wheel of momet of inertia $3I$ initially at rest is suddenly coupled to the same shaft. The resultant fractional loss in the kinetic energy of the system is:

• A. $0$
• B. $\frac{1}{4}$
• C. $\frac{3}{4}$
• D. $\frac{5}{6}$

Answer 1

Answer: (C)

By anglar momentum conservation
$\omega I+3I\times 0=4I{\omega }^{\prime }$
$\Rightarrow {\omega }^{\prime }=\frac{\omega }{4}$
$(KE{)}_i=\frac{1}{2}I{\omega }^2$
$(KE{)}_f=\frac{1}{2}\times (4I)\times {\left(\frac{\omega }{4}\right)}^2=\frac{I{\omega }^2}{8}$
$\Delta KE=\frac{3}{8}I{\omega }^2$
fractional loss $=\frac{\Delta KE}{K{E}_1}=\frac{\frac{3}{8}I{\omega }^2}{\frac{1}{2}I{\omega }^2}=\frac{3}{4}$