A block A of mass m is tied to a fixed point C on a horizontal table through a string passing round a mass less smooth pulley B.A force F is applied by the pulley B as shown in
(b) Find the accelerations of the pulley and mass A.

Explanation

To solve the problem of finding the accelerations of the pulley B and mass A, we can follow these steps:

### Step 1: Identify the Forces Acting on the Pulley
The pulley B is subjected to:
- An applied force \( F \) in the forward direction.
- Two tensions \( T \) acting backward due to the string on both sides of the pulley.

### Step 2: Write the Equation of Motion for the Pulley
Since the pulley is massless, we can apply Newton's second law:
\[ F - 2T = 0 \]
This implies:
\[ F = 2T \]
From this, we can express the tension \( T \):
\[ T = \frac{F}{2} \]

### Step 3: Analyze the Forces Acting on Mass A
For mass A, the only force acting on it is the tension \( T \) in the string. According to Newton's second law:
\[ T = m \cdot a_A \]
where \( a_A \) is the acceleration of mass A. Substituting the expression for \( T \):
\[ \frac{F}{2} = m \cdot a_A \]
Thus, we can solve for the acceleration of mass A:
\[ a_A = \frac{F}{2m} \]

### Step 4: Relate the Accelerations of the Pulley and Mass A
Next, we need to relate the acceleration of the pulley \( a_B \) to the acceleration of mass A. When the pulley moves a distance \( x \) forward, the length of the string on either side of the pulley changes. The relationship can be established as:
\[ \text{If the pulley moves forward by } x, \text{ then mass A moves backward by } 2x. \]
Thus, the acceleration of the pulley \( a_B \) is half of the acceleration of mass A:
\[ a_B = \frac{1}{2} a_A \]

### Step 5: Substitute the Expression for \( a_A \)
Substituting the expression for \( a_A \) into the equation for \( a_B \):
\[ a_B = \frac{1}{2} \left(\frac{F}{2m}\right) = \frac{F}{4m} \]

### Final Results
Thus, the accelerations of the pulley and mass A are:
- Acceleration of mass A:
\[ a_A = \frac{F}{2m} \]
- Acceleration of pulley B:
\[ a_B = \frac{F}{4m} \]