Vardaan Institute - Level 2 Solutions

1.

Two masses $m_1 = 3\text{ kg}$ and $m_2 = 5\text{ kg}$ connected by a light string pass over a frictionless pulley. Find system acceleration and string tension.

Sol: $a = \frac{m_2 - m_1}{m_1 + m_2} g = \frac{5 - 3}{8} g = \frac{g}{4} = 2.45\text{ m/s}^2$. $T = \frac{2 m_1 m_2}{m_1 + m_2} g = \frac{30}{8}(9.8) = 36.75\text{ N}$.

2.

A block slides down an inclined plane of angle $\theta=30^\circ$ with coefficient of kinetic friction $\mu_k = 0.2$. Find acceleration of the block.

Sol: $a = g(\sin\theta - \mu_k \cos\theta) = 9.8(0.5 - 0.2(0.866)) = 9.8(0.3268) = 3.20\text{ m/s}^2$.

3.

Derive maximum safe speed for a car negotiating a banked circular road of radius $R$ and banking angle $\theta$ with friction coefficient $\mu$.

Sol: $v_{max} = \sqrt{R g \left(\frac{\mu + \tan\theta}{1 - \mu \tan\theta}\right)}$.

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