A solid disc and a ring, both of radius 10 cm are placed on a horizontal table simultaneously, with initial angular speed equal to 10 π rad s–1. Which of the two will start to roll earlier ? The co-efficient of kinetic friction is uk = 0.2.

Engineering

Physics

Static and Kinetic Friction

Moment of Inertia

Rolling

Question

A solid disc and a ring, both of radius 10 cm are placed on a horizontal table simultaneously, with initial angular speed equal to 10 π rad s^–1. Which of the two will start to roll earlier ? The co-efficient of kinetic friction is u_k= 0.2.

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The motion of the two objects is caused by the frictional force acting on the objects = μ_kmg

Thus acceleration due to frictional force = μ_kg

Thus linear velocity attained in time t,

v = u + at = 0 + μ_kgt = μ_kgt

Also a torque would act due to friction causing a rotation about the center.

Thus τ = la

⇒ fr = la

Thus angular acceleration,

α = \frac{fr}{I} = \frac{μ_{k} mgr}{I}

Thus angular velocity attained after time t, ω = ω₀ – at

When rolling starts, v = rω

⟹ μ_{k} gt = r (ω_{0} - \frac{μ_{k} mgr}{I} t)

⟹ t = \frac{{rω}_{0}}{μ_{k} g + \frac{μ_{k} {mgr}^{2}}{I}}

Since l for ring is greater than that for solid disc, ring takes longer time to achieve pure rolling. Thus the solid disc starts to roll earlier.