Rolling Ring...

Classical Mechanics Level 2

A ring of radius r = 70 r = 70 cm and mass m = 500 m = 500 g rolls without slipping in a horizontal plane, of static and kinetic friction coefficients equal to μ = 0.4 \mu = 0.4 , when at time t = 0 t = 0 , it collides against a flat wall. Before the collision, the speed of the center of mass of the ring is s 0 = 3 s_0 = 3 m/s and the collision is instantaneous and perfectly elastic, such that it only results in the application of a horizontal impulse in the ring. Determine the time when the speed of the centre of the ring is constant.

Details and assumptions:

  • Acceleration of gravity g = 10 m / s 2 g = 10 \text{ m}/{\text{s}}^2
  • Moment of inertia of the ring I = m r 2 I = mr^2


The answer is 0.75.

A Former Brilliant Member by A Former Brilliant Member

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1 solution

After collision, the translational motion of the ring reverses direction, but rotational motion doesn't. So friction of plane on which it rolls will be in the direction opposite to that of motion. The linear acceleration of the centre of the ring is − μ g -\mu g and the angular acceleration is − μ g r \dfrac{-\mu g}{r} . The ring's speed will be constant when pure rolling prevails. This will happen when linear and angular speeds are related through v = r ω v=r\omega or s 0 = μ g t s_0=\mu gt or t = s 0 μ g = 0.75 t=\dfrac{s_0}{\mu g}=\boxed {0.75}

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