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 with friction, we need to analyze the forces acting on the disc and apply some principles of rotational motion and friction. 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 "ω" (omega). The friction between the disc and the surface will exert a torque that opposes the rotation of the disc. This frictional force is crucial in bringing the disc to a stop.

Calculating the Torque Due to Friction

The frictional force (F_f) acting on the disc can be calculated using the formula:

• F_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 (N = m * g, where "m" is the mass of the disc and "g" is the acceleration due to gravity).

Relating Torque to Angular Deceleration

The torque (τ) caused by the frictional force can be expressed as:

• τ = F_f * r

Substituting the expression for the frictional force, we get:

• τ = k * m * g * r

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

• τ = I * α

Where "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 Up the Equation

Now we can set the two expressions for torque equal to each other:

• k * m * g * r = (1/2) * m * r² * α

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

• α = (2 * k * g) / r

Finding the Time to Come to Rest

To find the time (t) it takes for the disc to come to rest, we can use the angular kinematics equation:

• ω_f = ω_i + α * t

Where:

• ω_f is the final angular velocity (0, since the disc comes to rest)
• ω_i is the initial angular velocity (ω)

Substituting the known values:

• 0 = ω + (-α) * t

Rearranging gives us:

• t = ω / α

Now, substituting the expression we found for α:

• t = ω / ((2 * k * g) / r)

Finally, simplifying this gives:

• t = (ω * r) / (2 * k * g)

Final Result

In summary, the time it takes for the disc to come to rest on a horizontal surface with a friction coefficient "k" is given by:

• 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 friction coefficient, and the acceleration due to gravity. Understanding these relationships helps in grasping the dynamics of rotating bodies and the effects of friction in real-world scenarios.