Question

A wheel is rolling on a plane surface. The speed of a particle on the highest point of the rim is 8 m/s. The speed of the particle on the rim of the wheel at the same level as the center of the wheel, will be:

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

Hint:

The speed of a point on the rim of a rolling wheel is the sum of the translational speed and the rotational speed. At the highest point, these speeds add up.

Answer 1

The Correct Option is A

The objective is to determine the speed of a particle on the rim of a wheel, which is rolling on a horizontal plane, specifically at a point level with the wheel's center. The solution proceeds as follows:

1. Given: The speed of a particle at the highest point of the wheel's rim is 8 m/s. For a wheel rolling without slipping, the speed at the highest point is twice the velocity of the wheel's center. Let \( V \) represent the velocity of the wheel's center. Therefore:
2. Next, we determine the speed of a rim particle located horizontally with respect to the wheel's center. At this position, the rotational velocity component is perpendicular to the translational velocity (center speed) of the wheel. The speed due to rotation at this point is equal to the center's speed, \( V = 4 \, \text{m/s} \). The wheel's translational speed is also \( 4 \, \text{m/s} \).
3. The resultant speed is calculated using the Pythagorean theorem, as the translational and rotational velocities are perpendicular:
4. Conclusion: The speed of the particle on the wheel's rim at the same vertical level as the wheel's center is \( 4\sqrt{2} \, \text{m/s} \).