Rotation 3 Question 10
16. The maximum value of $v _0$ for which the disk will roll without slipping is
(a) $\mu g \sqrt{\frac{M}{k}}$
(b) $\mu g \sqrt{\frac{M}{2 k}}$
(c) $\mu g \sqrt{\frac{3 M}{k}}$
(d) $\mu g \sqrt{\frac{5 M}{2 k}}$
Objective Questions II (One or more correct option)
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Answer:
Correct Answer: 16. (c)
Solution:
- In case of pure rolling, mechanical energy will remain conserved.
$$ \begin{array}{rlrl} & \therefore \frac{1}{2} M v _0^{2}+\frac{1}{2} \frac{1}{2} M R^{2} & \frac{v _0}{R} & =2 \frac{1}{2} k x^{2}{ } _{\max }^{2} \\ & \therefore \quad x _{\max } & =\sqrt{\frac{3 M}{4 k}} v _0 \\ & \text { As, } & =\frac{2 k x}{3} \\ & \therefore \quad F _{\max }=\mu M g=\frac{2 k x _{\max }}{3} & =\frac{2 k}{3} \sqrt{\frac{3 M}{4 k}} v _0 \\ & \therefore \quad v _0 & =\mu g \sqrt{\frac{3 M}{k}} \end{array} $$
