Two light strings of length 4 cm and 3 cm are tied to a bob of weight 500 gm. The free ends of the strings are tied to pegs in the same horizontal line and separated by 5 cm. The ratio of tension in the longer string to that in the shorter string is

Options

A. `4:3`
B. `3:4`
C. `4:5`
D. `5:4`

Answer

To solve the problem, we need to analyze the forces acting on the bob and apply the concept of equilibrium. Here’s a step-by-step solution:

### Step 1: Draw the Diagram
First, we need to visualize the situation. Draw a horizontal line representing the distance between the two pegs, which is 5 cm apart. At each end of this line, draw two strings: one of length 4 cm and the other of length 3 cm, both attached to a bob (weight = 500 gm) hanging downwards.

### Step 2: Identify the Tensions
Label the tension in the longer string (4 cm) as \( T_1 \) and the tension in the shorter string (3 cm) as \( T_2 \).

### Step 3: Resolve the Tensions into Components
Since the bob is in equilibrium, we can resolve the tensions into horizontal and vertical components. For the longer string:
- The horizontal component is \( T_1 \cos(\theta) \)
- The vertical component is \( T_1 \sin(\theta) \)

For the shorter string:
- The horizontal component is \( T_2 \cos(90^\circ - \theta) = T_2 \sin(\theta) \)
- The vertical component is \( T_2 \cos(\theta) \)

### Step 4: Apply Equilibrium Conditions
Since the bob is in equilibrium, the horizontal components of the tensions must balance each other:
\[ T_1 \cos(\theta) = T_2 \sin(\theta) \]

### Step 5: Find the Ratio of Tensions
From the equilibrium condition, we can express the ratio of the tensions:
\[ \frac{T_2}{T_1} = \frac{\cos(\theta)}{\sin(\theta)} = \cot(\theta) \]

### Step 6: Determine the Angles
Using the lengths of the strings and the horizontal distance between the pegs, we can find \( \cot(\theta) \):
- The vertical distance from the bob to the peg of the shorter string is 3 cm.
- The vertical distance from the bob to the peg of the longer string is 4 cm.
- The horizontal distance between the two pegs is 5 cm.

Using the right triangle formed by the shorter string:
\[ \cot(\theta) = \frac{\text{adjacent}}{\text{opposite}} = \frac{3 \text{ cm}}{4 \text{ cm}} \]

### Step 7: Final Calculation
Thus, we have:
\[ \frac{T_2}{T_1} = \cot(\theta) = \frac{3}{4} \]

### Step 8: Ratio of Tensions
To find the ratio of tension in the longer string to that in the shorter string:
\[ \frac{T_1}{T_2} = \frac{4}{3} \]

### Conclusion
The ratio of tension in the longer string to that in the shorter string is \( \frac{4}{3} \).