A solid circular disc m , R translates with v 0 and rotates with ω 0 = 2 v 0 R , at t = 0 . Because of friction, pure rolling starts at t = t 0 . The coefficient of friction is μ k . Find t 0 .

The frictional force provides the torque required for pure rolling motion(Torque is being calculated around an axis passing through the centre of the disc) τ = r × f ⇒ τ = f r sin θ here θ = 90 ° τ = f r We use the maximum value of friction possible here τ = μ k m g r Now torques can also be written as τ = I α So, - I α = μ k m g r .........(1) Here negative sign represents the retarding nature of the torque due to friction For a disc, the moment of inertia around an axis passing through its centre is m r 2 2 Using in equation (1) - m r 2 2 α = μ k m g r ⇒ α = - 2 μ k g r Let us assume, the velocity of the centre of disc when pure rolling has begun i.e. at time t o is v . Along the horizontal direction the only force acting is that of friction, thus: m a = f ⇒ m a = μ k m g ⇒ a = μ k g Here a Is constant and hence we can use v - u = a t ⇒ v - v o = μ k g t o ⇒ v = v o + μ k g t o Since there is a constant angular acceleration α = - 2 μ k g r We can use ω - ω o = α t ⇒ v r - ω o = - 2 μ k g r t o ⇒ v o + μ k g t o r - 2 v o r = - 2 μ k g r t o ⇒ v o + μ k g t o - 2 v o = - 2 μ k g t o ⇒ - v o = - 3 μ k g t o ⇒ t o = v o 3 μ k g

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A square plate is of mass $M$ and length of edge equal to $2 a$ . Its $M I$ about its centre of mass axis, perpendicular to its plane is equal to $I_{0}$ . Four identical disks are cut from the plane. The $M I$ of leftover square about same axis is

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Gamma Question Bank for Engineering Physics>Chapter 6 - System of Particles and Rotational Motion>EXERCISE 6.9>Q 54

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A square plate is of mass M and length of edge equal to 2a. Its MI about its centre of mass axis, perpendicular to its plane is equal to I0. Four identical disks are cut from the plane. The MI of leftover square about same axis is

Important Questions on System of Particles and Rotational Motion

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A square plate is of mass $M$ and length of edge equal to $2 a$ . Its $M I$ about its centre of mass axis, perpendicular to its plane is equal to $I_{0}$ . Four identical disks are cut from the plane. The $M I$ of leftover square about same axis is