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 - rad s -1 - Which of the two will start to roll earlier- The co-efficient of kinetic friction is - k -0-2-

Disc Radii of the ring and the disc, r = 10 cm = 0.1 m Initial angular speed, ω0 =10 π rad s–1 Coefficient of kinetic friction, μk = 0.2 Initial velocity of bot ...

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Standard XII

Physics

Pseudo Force

A solid disc ...

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 μ_k = 0.2.

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Solution

Disc

Radii of the ring and the disc, r = 10 cm = 0.1 m

Initial angular speed, ω₀=10 π rad s^–1

Coefficient of kinetic friction, μ_k = 0.2

Initial velocity of both the objects, u = 0

Motion of the two objects is caused by frictional force. As per Newton’s second law of motion, we have frictional force, f = ma

μ_kmg= ma

Where,

a = Acceleration produced in the objects

m = Mass

∴a = μ_kg … (i)

As per the first equation of motion, the final velocity of the objects can be obtained as:

v = u + at

= 0 + μ_kgt

= μ_kgt … (ii)

The torque applied by the frictional force will act in perpendicularly outward direction and cause reduction in the initial angular speed.

Torque, τ= –Iα

α = Angular acceleration

μ_xmgr = –Iα

Using the first equation of rotational motion to obtain the final angular speed:

Rolling starts when linear velocity, v = rω

Equating equations (ii) and (v), we get:

Since t_d > t_r, the disc will start rolling before the ring.

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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 μk = 0.2.

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 μ_k = 0.2.