Two identical uniform discs roll without slipping on two different surfaces AB and CD (see figure) starting at A and C with linear speeds v1 and v2, respectively, and always remain in contact with the surfaces. If they reach B and D with the same linear speed and v1 = 3 m/s, then v2 in m/s is(g = 10 m/s2)

Engineering

Physics

Rolling

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 v₁ and v₂, respectively, and always remain in contact with the surfaces. If they reach B and D with the same linear speed and v₁ = 3 m/s, then v₂ in m/s is
(g = 10 m/s²)

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$\frac{1}{2} {mv}_{1}^{2} + \frac{1}{2} \frac{{mr}^{2}}{2} \frac{v_{1}^{2}}{r^{2}} + mg (30) = \frac{1}{2} {mv}_{f}^{2} + \frac{1}{4} {mv}_{f}^{2} = \frac{3}{4} {mv}_{f}^{2} \Rightarrow for first ball$

$\frac{3}{4} {mv}_{1}^{2} + mg (30) = \frac{3}{4} {mv}_{f}^{2} = \frac{3}{4} {mv}_{2}^{2} + mg (27) \Rightarrow for both the balls$

$\frac{3}{4} {mv}_{2}^{2} = \frac{3}{4} {mv}_{1}^{2} + 3 mg \Rightarrow v_{2}^{2} = v_{1}^{2} + 4 g = 9 + 40 = 49$

v₂ = 7 m/sec.