If forces 12, 5 and 13 units weight balance at a point two of them are inclined at

Options

A. `30^(@)`
B. `45^(@)`
C. `60^(@)`
D. `90^(@)`

Answer

To solve the problem of balancing three forces of 12, 5, and 13 units at a point, where two of them are inclined, we can follow these steps:

### Step 1: Understand the Problem
We have three forces:
- F1 = 12 units
- F2 = 5 units
- F3 = 13 units

Two of these forces are inclined, and they balance each other along with the third force.

### Step 2: Identify the Forces and Angles
Assume that:
- F1 (12 units) is acting at an angle θ1 to the horizontal.
- F2 (5 units) is acting at an angle θ2 to the horizontal.
- F3 (13 units) is acting vertically (either upwards or downwards).

### Step 3: Resolve the Forces into Components
We can resolve the inclined forces into their horizontal and vertical components.

For F1:
- Horizontal component (F1x) = F1 * cos(θ1) = 12 * cos(θ1)
- Vertical component (F1y) = F1 * sin(θ1) = 12 * sin(θ1)

For F2:
- Horizontal component (F2x) = F2 * cos(θ2) = 5 * cos(θ2)
- Vertical component (F2y) = F2 * sin(θ2) = 5 * sin(θ2)

### Step 4: Set Up the Equilibrium Conditions
For the system to be in equilibrium, the sum of the horizontal components must equal zero, and the sum of the vertical components must also equal zero.

1. **Horizontal equilibrium**:
\[ F1x + F2x = F3x \]
Since F3 is vertical, F3x = 0, so:
\[ 12 \cos(θ1) + 5 \cos(θ2) = 0 \]

2. **Vertical equilibrium**:
\[ F1y + F2y = F3y \]
Assuming F3 acts downwards, we have:
\[ 12 \sin(θ1) + 5 \sin(θ2) = 13 \]

### Step 5: Solve the Equations
From the horizontal equilibrium equation:
\[ 12 \cos(θ1) + 5 \cos(θ2) = 0 \implies \cos(θ2) = -\frac{12}{5} \cos(θ1) \]
This indicates that θ2 must be such that the cosine value is negative, which is possible in the second quadrant.

From the vertical equilibrium equation:
\[ 12 \sin(θ1) + 5 \sin(θ2) = 13 \]

### Step 6: Substitute and Solve for Angles
Using the relationship obtained from the horizontal equilibrium, we can substitute cos(θ2) into the vertical equation and solve for the angles θ1 and θ2.

### Step 7: Check the Solution
Once we find the angles, we should check if both equilibrium conditions are satisfied.