A mass of `2.9 kg` is suspended from a string of length `50 cm` and is at rest . Another body of mass `100 gm` which is moving horizontal with a velocity of `150 m//s` strikes it . After striking the two bodies combine together . Tension in the string , when it is at an angle of `60^(@)` with the velocity is : `g = 10 m//s^(2)`

Options

A. `135 N`
B. `125 N`
C. `140 N`
D. `90 N`

Explanation

To solve the problem step by step, we will follow the principles of momentum conservation and forces acting on the pendulum after the collision.

### Step 1: Understand the system
We have two masses:
- Mass of the pendulum (M) = 2.9 kg
- Mass of the moving body (m) = 100 g = 0.1 kg
- Length of the string (L) = 50 cm = 0.5 m
- Angle (θ) = 60 degrees
- Gravitational acceleration (g) = 10 m/s²

### Step 2: Calculate the initial momentum
The initial momentum of the system before the collision is given by the moving mass:
\[ P_{initial} = m \cdot u + M \cdot 0 = 0.1 \cdot 150 + 2.9 \cdot 0 = 15 \, \text{kg m/s} \]

### Step 3: Apply conservation of momentum
After the collision, the two masses combine and move together with a common velocity (v). According to the conservation of momentum:
\[ m \cdot u = (m + M) \cdot v \]
Substituting the known values:
\[ 0.1 \cdot 150 = (0.1 + 2.9) \cdot v \]
\[ 15 = 3 \cdot v \]
\[ v = \frac{15}{3} = 5 \, \text{m/s} \]

### Step 4: Analyze forces acting on the pendulum
When the pendulum is at an angle of 60 degrees, the forces acting on it are:
1. The tension (T) in the string.
2. The weight of the combined mass (m + M) acting downwards, which is \( (m + M)g \).

### Step 5: Resolve forces
The weight can be resolved into two components:
- The component along the direction of the string: \( (m + M)g \cos(θ) \)
- The component perpendicular to the string, which provides the centripetal force: \( (m + M)g \sin(θ) \)

### Step 6: Write the equation for tension
Using the balance of forces in the vertical direction:
\[ T - (m + M)g \cos(θ) = \frac{(m + M)v^2}{L} \]
Where \( \frac{(m + M)v^2}{L} \) is the centripetal force required to keep the mass moving in a circular path.

### Step 7: Substitute known values
Substituting the values:
- \( m + M = 3 \, \text{kg} \)
- \( g = 10 \, \text{m/s}^2 \)
- \( v = 5 \, \text{m/s} \)
- \( L = 0.5 \, \text{m} \)
- \( θ = 60^\circ \) (cos(60°) = 0.5)

The equation becomes:
\[ T - 3 \cdot 10 \cdot 0.5 = \frac{3 \cdot 5^2}{0.5} \]
\[ T - 15 = \frac{3 \cdot 25}{0.5} \]
\[ T - 15 = 150 \]
\[ T = 150 + 15 = 165 \, \text{N} \]

### Final Answer
The tension in the string when it is at an angle of 60 degrees with the vertical is **165 N**.