A cart consists of a body of mass m and two wheels, each of mass m and radius R. The cart is attached to a spring of constant k . The other end of the spring is fixed to a wall as shown in figure. If time period of oscillation is $β \times 2 π \frac{m}{k}$ then find the value of $β$ . (Assume pure rolling)

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TestHub · Accepted Answer

KE of the system: body (mass m) translates; each wheel (mass m, I = ½mR²) translates and rotates with ω = v/R (pure rolling).

$K E = \frac{1}{2} m v^{2} + 2 (\frac{1}{2} m v^{2} + \frac{1}{2} \times \frac{1}{2} m R^{2} \times \frac{v ^{2}}{R ^{2}}) = \frac{1}{2} (4 m) v^{2}$

Effective mass is 4m, so: $T = 2 π \frac{4 m}{k} = 2 \times 2 π \frac{m}{k}$

Comparing with β × 2π√(m/k): β = 2

Answer: 2

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