A steel wire can withstand a load up to `2940N`. A load of `150 kg` is suspended from a rigid support. The maximum angle through which the wire can be displaced from the mean position, so that the wire does not break when the load pass through the position of equilibrium, is

Options

A. `30^@`
B. `60^@`
C. `80^@`
D. `85^@`

Answer

To solve the problem step by step, we need to analyze the forces acting on the steel wire when a load is suspended from it and it is displaced at an angle θ from the vertical.

### Step 1: Understand the forces acting on the wire
When the wire is displaced by an angle θ, two main forces act on the mass:
1. The gravitational force (weight) acting downward, which is given by \( mg \).
2. The tension \( T \) in the wire, which can be resolved into two components:
- \( T \cos \theta \) acting vertically upward.
- \( T \sin \theta \) acting horizontally.

### Step 2: Set up the equilibrium condition
At the equilibrium position, the vertical component of the tension must balance the weight of the mass. Therefore, we can write the equation:
\[ T \cos \theta = mg \]

### Step 3: Substitute known values
Given:
- The maximum load the wire can withstand is \( T_{max} = 2940 \, \text{N} \).
- The mass \( m = 150 \, \text{kg} \).
- The acceleration due to gravity \( g = 9.8 \, \text{m/s}^2 \).

Calculate the weight \( mg \):
\[ mg = 150 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 1470 \, \text{N} \]

### Step 4: Substitute into the equilibrium equation
Now substituting \( mg \) into the equilibrium equation:
\[ T \cos \theta = 1470 \, \text{N} \]

### Step 5: Relate tension to maximum tension
Since we want to ensure that the tension does not exceed the maximum allowable tension, we set \( T = 2940 \, \text{N} \):
\[ 2940 \cos \theta = 1470 \]

### Step 6: Solve for \( \cos \theta \)
Rearranging the equation gives:
\[ \cos \theta = \frac{1470}{2940} = \frac{1}{2} \]

### Step 7: Find the angle \( \theta \)
Now, we can find \( \theta \) using the cosine inverse:
\[ \theta = \cos^{-1}\left(\frac{1}{2}\right) \]
This gives:
\[ \theta = 60^\circ \]

### Conclusion
The maximum angle through which the wire can be displaced from the mean position without breaking is \( 60^\circ \).