In the figure shown coefficient of friction between 8 kg block and surface is zero and coefficient of friction ( both static and kinetic ) between the blocks is 0.4. A force of 20 N is applied on 8 kg block as shown . The acceleration of 4 kg mass is [ g = `10 m//s^(2)`]

A. ` 4 m//s^(2)`
B. `0.5 m//s^(2)`
C. `5/3 m//s^(2)`
D. `3/5 m//s^(2)`

Options

A. ` 4 m//s^(2)`
B. `0.5 m//s^(2)`
C. `5/3 m//s^(2)`
D. `3/5 m//s^(2)`

Answer

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 forces acting on the 4 kg block
The only force acting on the 4 kg block (let's call it Block B) is the frictional force exerted by the 8 kg block (Block A). The frictional force can be calculated using the formula:
\[ F_{\text{friction}} = \mu \cdot m \cdot g \]
where:
- \( \mu = 0.4 \) (coefficient of friction between the blocks),
- \( m = 4 \, \text{kg} \) (mass of Block B),
- \( g = 10 \, \text{m/s}^2 \) (acceleration due to gravity).

### Step 2: Calculate the frictional force
Substituting the values into the formula:
\[ F_{\text{friction}} = 0.4 \cdot 4 \, \text{kg} \cdot 10 \, \text{m/s}^2 \]
\[ F_{\text{friction}} = 0.4 \cdot 40 \]
\[ F_{\text{friction}} = 16 \, \text{N} \]

### Step 3: Determine the maximum acceleration of the 4 kg block
Using Newton's second law \( F = m \cdot a \), we can find the acceleration of Block B:
\[ F_{\text{friction}} = m \cdot a \]
\[ 16 \, \text{N} = 4 \, \text{kg} \cdot a \]
To find \( a \):
\[ a = \frac{16 \, \text{N}}{4 \, \text{kg}} \]
\[ a = 4 \, \text{m/s}^2 \]

### Step 4: Analyze the system when both blocks move together
Next, we consider the scenario where both blocks move together. The total mass of the system is:
\[ m_{\text{total}} = 8 \, \text{kg} + 4 \, \text{kg} = 12 \, \text{kg} \]
The total force applied on the system is 20 N. According to Newton's second law:
\[ F_{\text{net}} = m_{\text{total}} \cdot a \]
Thus:
\[ 20 \, \text{N} = 12 \, \text{kg} \cdot a \]
To find \( a \):
\[ a = \frac{20 \, \text{N}}{12 \, \text{kg}} \]
\[ a = \frac{5}{3} \, \text{m/s}^2 \]

### Conclusion
The acceleration of the 4 kg mass when both blocks move together is:
\[ a = \frac{5}{3} \, \text{m/s}^2 \]

### Final Answer
The correct option is C: \( \frac{5}{3} \, \text{m/s}^2 \).

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