Question

A rigid circular disc of radius \(r\) (in m) is rolling without slipping on a flat surface as shown in the figure below. The angular velocity of the disc is \(\omega\) (in rad/ssuperscript{-1}). The velocities (in m/ssuperscript{-1}) at points 0 and A, respectively, are:

 

• A. \(r \omega \hat{i}\) and \(0 \hat{i}\)
• B. \(-r \omega \hat{i}\) and \(0 \hat{i}\)
• C. \(-r \omega \hat{i}\) and \(-r \omega \hat{i}\)
• D. \(r \omega \hat{i}\) and \(r \omega \hat{i}\)

Hint:

In rolling motion without slipping, the velocity at the point of contact with the surface is always zero, while the center of the disc has a velocity proportional to the angular velocity and the radius.

Answer 1

The Correct Option is A

In this problem, a rigid circular disc of radius \( r \) is rolling without slipping on a flat surface. The angular velocity of the disc is given by \( \omega \) (in rad s⁻¹), and we are asked to find the velocities at points 0 (the center of the disc) and A (the point of contact with the surface).

Step 1: Velocity at point 0 (center of the disc)
The center of the disc (point 0) is moving with a velocity \( v_0 \) given by:
\[ v_0 = r \omega \]
where \( r \) is the radius of the disc and \( \omega \) is the angular velocity. The velocity at point 0 is directed along the horizontal axis \( \hat{i} \).

Thus, the velocity at point 0 is \( r \omega \hat{i} \).

Step 2: Velocity at point A (point of contact)
The point A is the point of contact with the flat surface. Since the disc is rolling without slipping, the velocity at this point is zero relative to the surface. This means:
\[ v_A = 0 \]
Thus, the velocity at point A is \( 0 \hat{i} \).

Step 3: Combine the results
- Velocity at point 0: \( r \omega \hat{i} \)
- Velocity at point A: \( 0 \hat{i} \)

Thus, the correct answer is (A) \( r \omega \hat{i} \) and \( 0 \hat{i} \).