A wheel of radius 2 m rolls on the ground with uniform velocity `4 m s^(-1)`. . The relative acceleration of the topmost point of the wheel with respect to the bottom - most point of the wheel is

Options

A. `8ms^(-2)`
B. `16ms^(-2)`
C. `4ms^(-2)`
D. `32ms^(-2)`

Answer

To solve the problem of finding the relative acceleration of the topmost point (Q) of a rolling wheel with respect to the bottommost point (P), we can follow these steps:

### Step 1: Understand the motion of the wheel
The wheel is rolling with a uniform velocity of \( V = 4 \, \text{m/s} \) and has a radius \( R = 2 \, \text{m} \).

### Step 2: Calculate the angular velocity (\( \omega \))
Using the relationship between linear velocity and angular velocity:
\[ \omega = \frac{V}{R} \]
Substituting the values:
\[ \omega = \frac{4 \, \text{m/s}}{2 \, \text{m}} = 2 \, \text{rad/s} \]

### Step 3: Determine the accelerations at points P and Q
- The bottommost point (P) of the wheel is in contact with the ground and has zero acceleration in the vertical direction because it is not moving.
- The topmost point (Q) of the wheel is moving in a circular path and experiences centripetal acceleration.

### Step 4: Calculate the centripetal acceleration at point Q
The centripetal acceleration (\( a_c \)) is given by:
\[ a_c = \omega^2 R \]
Substituting the values:
\[ a_c = (2 \, \text{rad/s})^2 \times 2 \, \text{m} = 4 \times 2 = 8 \, \text{m/s}^2 \]

### Step 5: Calculate the total acceleration of point Q
The total acceleration of point Q consists of the centripetal acceleration directed towards the center of the wheel and the tangential acceleration, which is zero since the wheel rolls with uniform velocity. Therefore, the total acceleration of Q is:
\[ a_Q = a_c = 8 \, \text{m/s}^2 \]

### Step 6: Determine the relative acceleration of Q with respect to P
Since point P has zero acceleration, the relative acceleration of Q with respect to P is simply the acceleration of Q:
\[ a_{Q/P} = a_Q - a_P = 8 \, \text{m/s}^2 - 0 = 8 \, \text{m/s}^2 \]

### Final Answer
The relative acceleration of the topmost point of the wheel with respect to the bottommost point of the wheel is:
\[ \boxed{8 \, \text{m/s}^2} \]