Question:
Initial angular velocity of a circular disc of mass $M$ is $\omega_{1}$. Then two small spheres of mass $m$ are attached gently to two diametrically opposite points on the edge of the disc. What is the final angular velocity of the disc?
Initial angular velocity of a circular disc of mass $M$ is $\omega_{1}$. Then two small spheres of mass $m$ are attached gently to two diametrically opposite points on the edge of the disc. What is the final angular velocity of the disc?
Updated On: Jul 2, 2022
- $\left(\frac{M+m}{M}\right) \omega_{1}$
- $\left(\frac{M+m}{m}\right) \omega_{1}$
- $\left(\frac{M}{M+4 m}\right) \omega_{1}$
- $\left(\frac{M}{M+2 m}\right) \omega_{1}$
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The Correct Option is C
Solution and Explanation
Conservation of angular momentum gives
$\frac{1}{2} M R^{2} \omega_{1}=\left(\frac{1}{2} M R^{2}+2 m R^{2}\right) \omega_{2}$
$\Rightarrow \frac{1}{2} M R^{2} \omega_{1}=\frac{1}{2} R^{2}(M+4 m) \omega_{2}$
$\therefore \omega_{2}=\left(\frac{M}{M+4 m}\right) \omega_{1}$
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Concepts Used:
System of Particles and Rotational Motion
- The system of particles refers to the extended body which is considered a rigid body most of the time for simple or easy understanding. A rigid body is a body with a perfectly definite and unchangeable shape.
- The distance between the pair of particles in such a body does not replace or alter. Rotational motion can be described as the motion of a rigid body originates in such a manner that all of its particles move in a circle about an axis with a common angular velocity.
- The few common examples of rotational motion are the motion of the blade of a windmill and periodic motion.