First slide

Rotational motion

Question

A wheel is rotating freely with an angular speed $ω$ 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 moment of inertia $3 I$ initially at rest is suddenly coupled to the same shaft. The resultant fractional loss in the kinetic energy of the system is:

Moderate

A

0
B

$\frac{1}{4}$
C

$\frac{5}{6}$
D

$\frac{3}{4}$

Solution

$C o n s e r v a t i o n o f A n g u l a r M o m e n t u m, \Rightarrow I ω + (3 I) (0) = I ω^{1} + 3 I ω^{1} \begin{array}{l} \Rightarrow \frac{ω}{4} = ω^{1} \\ \Rightarrow L o s s i n K i n e t i c e n e r g y = K_{i} - K_{f} = \frac{1}{2} {Iω}^{2} - \frac{1}{2} 4I {(\frac{ω}{4})}^{2} = \frac{3}{8} {Iω}^{2} \\ \Rightarrow F r a c t i o n a l l o s s = \frac{K_{i} - K_{f}}{K_{i}} = \frac{\frac{3}{8} {Iω}^{2}}{\frac{1}{2} {Iω}^{2}} = \frac{3}{4} \end{array}$

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