Question

A wheel is rolling on a plane surface. A point on the rim of the wheel at the same level as the Centre has a speed of 4m/s. The speed of the Centre of the wheel is:

• A. 4m/s
• B. 0
• C. 2m/s
• D. 8m/s
• E. 4\(\sqrt{2}\)m/s

Answer 1

The Correct Option is C

Given:

• A wheel is rolling on a plane surface.
• A point on the rim (at the same height as the center) has speed \( v_{\text{rim}} = 4 \, \text{m/s} \).

Step 1: Understand Rolling Motion

For a rolling wheel:

• The center moves with speed \( v_{\text{center}} \).
• Due to rotation, points on the rim have additional tangential speed \( v_{\text{tangential}} = R\omega \), where \( R \) is the radius and \( \omega \) is angular velocity.
• In pure rolling, \( v_{\text{center}} = R\omega \).

Step 2: Analyze the Given Point

At the same height as the center, the point's total velocity is the vector sum of:

1. Translational velocity of the center: \( v_{\text{center}} \) (horizontal).
2. Tangential velocity due to rotation: \( v_{\text{tangential}} = R\omega = v_{\text{center}} \) (horizontal, opposite to translation).

Thus, the net speed at this point is:

\[ v_{\text{rim}} = v_{\text{center}} - (-v_{\text{center}}) = 2v_{\text{center}} \]

(The negative sign accounts for the opposite direction of rotation at this point.)

Step 3: Solve for \( v_{\text{center}} \)

Given \( v_{\text{rim}} = 4 \, \text{m/s} \):

\[ 4 = 2v_{\text{center}} \implies v_{\text{center}} = 2 \, \text{m/s} \]

Conclusion:

Based on standard rolling motion analysis, the center speed should be 2 m/s.

Answer 2

1. Understand the motion of a rolling wheel:

For a wheel rolling without slipping, the instantaneous velocity at the point of contact with the ground is zero. The velocity at the top of the wheel is twice the velocity of the center of the wheel. The velocity of any point on the rim is the vector sum of the translational velocity of the center and the rotational velocity about the center.

2. Analyze the given information:

We're given that the speed of a point on the rim at the same level as the center is 4 m/s. This point is on the horizontal diameter of the wheel and moving forward due to both rotation and translation.

3. Define variables:

Let v_c be the speed of the center of the wheel.

Let v_r be the rotational speed of a point on the rim about the center. Since the wheel rolls without slipping, v_r is also equal to v_c.

4. Calculate the speed of the center:

At the point on the rim level with the center, the translational velocity of the center (v_c) and the rotational velocity (v_r) are in the same direction (horizontally forward). Therefore, the speed of this point is the sum of these two velocities:

\[v_{rim} = v_c + v_r\]

Since \(v_r = v_c\), we have:

\[4 \, m/s = v_c + v_c = 2v_c\]

\[v_c = \frac{4 \, m/s}{2} = 2 \, m/s\]