Two identical uniform discs roll without slipping on two different surfaces $A B$ and $C D$ (see figure) starting at $A$ and $C$ with linear speeds $v_{1}$ and $v_{2}$ , 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 ( $g = 10 {m/s}^{2}$ ) [2015]

Question

Smart Achievers · Accepted Answer

(7)

$K . E_{total} of a rolling disc = K . E_{trans} + K . E_{rot}$

$= \frac{1}{2} m v^{2} + \frac{1}{2} I ω^{2} = \frac{1}{2} m v^{2} + \frac{1}{2} (\frac{1}{2} m R^{2}) (\frac{v^{2}}{R^{2}})$

$∴$ $K . E_{total} = \frac{3}{4} m v^{2}$

$For surface A B$

$K . E_{i} + loss in gravitational potential energy = K . E_{f}$

$\frac{3}{4} m {(3)}^{2} + m g (30) = \frac{3}{4} m V_{B}^{2} ...(i)$

$For surface C D$

$\frac{3}{4} m {(v_{2})}^{2} + m g (27) = \frac{3}{4} m V_{D}^{2} ...(ii)$

$∵$ $V_{B} = V_{D}$ $∴$ $from eq. (i) and (ii)$

$\frac{3}{4} m {(3)}^{2} + m g \times 30 = \frac{3}{4} m {(v_{2})}^{2} + m g \times 27$

$∴$ $v_{2} = 7 m/s$

Solution

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