A uniform disc of radius R rotating with angular velocity w is carefully placed on a horizontal surface having friction coefficient k . After how much time the disc will come to rest?

To determine how long it takes for a uniform disc to come to rest after being placed on a horizontal surface, we need to consider the forces acting on the disc and the role of friction. The friction between the disc and the surface will exert a torque that opposes the rotation of the disc, ultimately bringing it to a stop. Let's break this down step by step.

Understanding the Forces at Play

When the disc is placed on the surface, it has an initial angular velocity, denoted as "ω" (omega). The frictional force acts at the edge of the disc, providing a torque that slows down its rotation. The frictional force can be calculated using the formula:

• Frictional Force (F) = k * N

Here, "k" is the coefficient of friction, and "N" is the normal force. For a disc resting on a horizontal surface, the normal force is equal to the weight of the disc, which can be expressed as:

• N = m * g

where "m" is the mass of the disc and "g" is the acceleration due to gravity.

Calculating the Torque

The torque (τ) caused by the frictional force can be calculated using the formula:

• Torque (τ) = F * R

Substituting the expression for the frictional force, we get:

• τ = (k * m * g) * R

Relating Torque to Angular Deceleration

According to Newton's second law for rotation, the torque is also related to the angular deceleration (α) of the disc:

• τ = I * α

Here, "I" is the moment of inertia of the disc. For a uniform disc, the moment of inertia is given by:

• I = (1/2) * m * R²

Setting the two expressions for torque equal to each other, we have:

• (k * m * g) * R = (1/2) * m * R² * α

Solving for Angular Deceleration

We can simplify this equation by canceling out the mass "m" (assuming it is not zero) and rearranging to find the angular deceleration:

• α = (2 * k * g) / R

Finding the Time to Come to Rest

Now that we have the angular deceleration, we can use the kinematic equation for rotational motion to find the time (t) it takes for the disc to come to rest. The equation we will use is:

• ω_f = ω_i + α * t

Here, "ω_f" is the final angular velocity (0, since the disc comes to rest), and "ω_i" is the initial angular velocity (ω). Plugging in the values, we get:

• 0 = ω - (2 * k * g / R) * t

Rearranging this gives us:

• t = (ω * R) / (2 * k * g)

Final Expression

Thus, the time it takes for the disc to come to rest after being placed on the surface is:

• t = (ω * R) / (2 * k * g)

This formula allows you to calculate the stopping time based on the initial angular velocity, the radius of the disc, the coefficient of friction, and the acceleration due to gravity. By plugging in the appropriate values, you can find the exact time it takes for the disc to come to a complete stop.