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Pure Rolling

Two identical...

Question

Two identical uniform discs roll without slipping on two different surfaces AB and CD (see figure) starting at A and C with linear speeds $v_{1} a n d v_{2},$ respectively, and always remain in contact with the surfaces. If they reach B and D with the same linear speed and $v_{1} = 3$ m/s, then $v_{2}$ in m/s is ___. $(g = 10 m / s^{2})$

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Solution

Kinetic energy of a purelyrolling disc
$= \frac{1}{2} m V^{2} + \frac{1}{2} (\frac{m}{2}) V^{2} = \frac{3}{4} m V^{2}$
Using conservation of energy on both discs and equating their final energies.
$\frac{3}{4} m V_{1}^{2} + m g . 30 = \frac{3}{4} m V_{2}^{2} + m g . 27$
$3 m g = \frac{3}{4} m (V_{2}^{2} - V_{1}^{2})$
$4 g = V_{2}^{2} - 9$
$V_{2}^{2} = 49$
$V_{2} = 7$

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