A wheel is rolling along the ground with a speed of `2ms^(-1)` The magnitude of the velocity of the points at the extremities of the horizontal diameter of the wheel is equal to

Options

A. `2sqrt(10)ms^(-1)`
B. `2sqrt(3)ms^(-1)`
C. `2sqrt(2)ms^(-1)`
D. `2ms^(-1)`

Explanation

To solve the problem of finding the magnitude of the velocity of the points at the extremities of the horizontal diameter of a wheel rolling on the ground with a speed of \(2 \, \text{m/s}\), we can follow these steps:

### Step 1: Understand the Motion of the Wheel
When a wheel rolls without slipping, every point on the wheel has a velocity that can be broken down into two components:
1. The translational velocity of the center of mass of the wheel.
2. The rotational velocity due to the wheel's rotation about its center.

### Step 2: Identify the Components of Velocity
- The translational velocity \(v\) of the wheel's center is given as \(2 \, \text{m/s}\).
- The angular velocity \(\omega\) can be related to the translational velocity by the equation:
\[ v = r \omega \]
where \(r\) is the radius of the wheel.

### Step 3: Calculate the Velocity at the Extremities
For a point at the top of the wheel:
- The velocity due to translation is \(2 \, \text{m/s}\) (in the direction of motion).
- The velocity due to rotation (upward) is also \(r \omega = v = 2 \, \text{m/s}\).

Thus, the total velocity at the top of the wheel is:
\[ v_{\text{top}} = v + r \omega = 2 + 2 = 4 \, \text{m/s} \]

For a point at the bottom of the wheel:
- The velocity due to translation is \(2 \, \text{m/s}\) (in the direction of motion).
- The velocity due to rotation (downward) is \(r \omega = v = 2 \, \text{m/s}\) but in the opposite direction.

Thus, the total velocity at the bottom of the wheel is:
\[ v_{\text{bottom}} = v - r \omega = 2 - 2 = 0 \, \text{m/s} \]

### Step 4: Conclusion
The magnitude of the velocity of the points at the extremities of the horizontal diameter of the wheel is:
- Top point: \(4 \, \text{m/s}\)
- Bottom point: \(0 \, \text{m/s}\)

Thus, the magnitude of the velocity of the points at the extremities of the horizontal diameter is \(4 \, \text{m/s}\) at the top and \(0 \, \text{m/s}\) at the bottom.

### Final Answer
The magnitude of the velocity of the points at the extremities of the horizontal diameter of the wheel is \(4 \, \text{m/s}\) at the top and \(0 \, \text{m/s}\) at the bottom.

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