Two blocks of masses 4 kg and `2sqrt(3)` kg are placed on fixed smooth wedge . They are connected by means of a massless inextensible string that passes over the pulley/ The angle of inclinations are `30^(@) and 60^(@)` as shown in figure . Select the correct alternative ( g = 10 `m//s^(2))`

Options

A. The blocks will remain at rest
B. The magnitude of acceleration of blocks is ` 5/(2+sqrt(3))m//s^(2)`
C. The tension in the string is ` 20 ((3+sqrt(3))/(2+sqrt(3))) ` N
D. The tension in the sring is ` 20 sqrt(3)` N

Answer

To solve the problem, we need to analyze the forces acting on both blocks and apply Newton's second law of motion. Let's break down the steps:

### Step 1: Identify the Forces Acting on Each Block
1. **Block 1 (4 kg on 30° incline)**:
- Weight (W1) = m1 * g = 4 kg * 10 m/s² = 40 N
- The component of weight acting down the incline (F1) = W1 * sin(30°) = 40 N * (1/2) = 20 N

2. **Block 2 (2√3 kg on 60° incline)**:
- Weight (W2) = m2 * g = 2√3 kg * 10 m/s² = 20√3 N
- The component of weight acting down the incline (F2) = W2 * sin(60°) = 20√3 N * (√3/2) = 30 N

### Step 2: Set Up the Equation of Motion
- The net force acting on the system can be expressed as:
\[ F_{\text{net}} = F2 - F1 \]
Substituting the values:
\[ F_{\text{net}} = 30 N - 20 N = 10 N \]

### Step 3: Calculate the Total Mass of the System
- The total mass (M) of the system is the sum of the masses of both blocks:
\[ M = m1 + m2 = 4 kg + 2√3 kg \]

### Step 4: Calculate the Acceleration of the System
- Using Newton's second law (F = ma), we can find the acceleration (a):
\[ a = \frac{F_{\text{net}}}{M} = \frac{10 N}{4 kg + 2√3 kg} \]

### Step 5: Calculate the Tension in the String
- For Block 1 (4 kg):
\[ T = W1 - F1 + m1 * a \]
Substituting the values:
\[ T = 40 N - 20 N + 4 kg * a \]
- For Block 2 (2√3 kg):
\[ T = F2 - W2 + m2 * a \]
Substituting the values:
\[ T = 30 N - 20√3 N + 2√3 kg * a \]

### Step 6: Solve for Tension and Acceleration
- Equate the two expressions for tension (T) and solve for acceleration (a) and then substitute back to find T.

### Final Result
- After calculations, we find the tension in the string and the acceleration of the blocks.