A block A of mass 4 kg is placed on another block B of mass 5 kg , and the block B rests on a smooth horizontal table . For sliding block A on B , a horizontal force of 12 N is required to be applied on A How much maximum force can be applied on B so that both A and B move togrther ? Also find out acceleration produced by this force .

Options

A. (a)15 N
B. (b)25 N
C. (c)27 N
D. (d)48 N

Explanation

To solve the problem step by step, we will analyze the forces acting on the blocks and apply Newton's laws of motion.

### Step 1: Identify the masses and forces
- Mass of block A (m1) = 4 kg
- Mass of block B (m2) = 5 kg
- Force required to slide block A on B (F) = 12 N

### Step 2: Determine the maximum static friction force
The maximum static friction force (f_max) that can act between blocks A and B is what allows them to move together without slipping. This can be calculated using the formula:
\[ f_{\text{max}} = \mu_s \cdot N \]
where \( N \) is the normal force acting on block A due to its weight. Since block A is resting on block B, the normal force is equal to the weight of block A:
\[ N = m_1 \cdot g = 4 \, \text{kg} \cdot 9.8 \, \text{m/s}^2 = 39.2 \, \text{N} \]

### Step 3: Calculate the maximum static friction force
Assuming the coefficient of static friction (\( \mu_s \)) is sufficient to prevent slipping, we can set:
\[ f_{\text{max}} = 12 \, \text{N} \]
This means that the maximum force that can be applied on block B without causing A to slip off is equal to the maximum static friction force.

### Step 4: Calculate the total mass when both blocks move together
When both blocks move together, the total mass \( M \) is:
\[ M = m_1 + m_2 = 4 \, \text{kg} + 5 \, \text{kg} = 9 \, \text{kg} \]

### Step 5: Calculate the maximum force that can be applied on block B
The maximum force \( F_B \) that can be applied on block B, so that both A and B move together, is equal to the maximum static friction force:
\[ F_B = f_{\text{max}} = 12 \, \text{N} \]

### Step 6: Calculate the acceleration produced by this force
Using Newton's second law, the acceleration \( a \) of the system when the force \( F_B \) is applied is given by:
\[ F_B = M \cdot a \]
Rearranging gives:
\[ a = \frac{F_B}{M} = \frac{12 \, \text{N}}{9 \, \text{kg}} \approx 1.33 \, \text{m/s}^2 \]

### Final Answers
- The maximum force that can be applied on block B so that both A and B move together is **12 N**.
- The acceleration produced by this force is approximately **1.33 m/s²**.