Rotational Motion

JEE Main level, single correct, medium

217 of 280

Question 217 of 280

A wheel is rotating freely with an angular speed ω\omega on a shaft. The moment of inertia of the wheel is II and the moment of inertia of the shaft is negligible. Another wheel of moment of inertia 3I3I initially at rest is suddenly coupled to the same shaft. The resultant fractional loss in the kinetic energy of the system is:

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Answer: (D) 34\dfrac{3}{4}

From angular momentum conservation
Iω+0=IωC+3IωCI\omega +0=I{\omega }_{C}+3I{\omega }_{C}
ωc=ω4\omega c=\dfrac{\omega }{4}
Loss of kinetic energy =12Iω2−12(I+3I)(ω4)2=\dfrac{1}{2}I{\omega }^{2}-\dfrac{1}{2}(I+3I){\left(\dfrac{\omega }{4}\right)}^{2}
=12Iω2−12Iω24=\dfrac{1}{2}I{\omega }^{2}-\dfrac{1}{2}I\dfrac{{\omega }^{2}}{4}
=38Iω2=\dfrac{3}{8}I{\omega }^{2}
Fractional loss =34=\dfrac{3}{4}

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