A uniform metre stick of mass 200 g is suspended from the ceiling through two vertical strings of equal lengths fixed at the ends. A small object of mass 20 g is placed on the stick at a distance of 70 cm fomd the left end. Find the tensions in the two strings.

Explanation

To solve the problem of finding the tensions in the two strings supporting the uniform metre stick with a small object placed on it, we can follow these steps:

### Step 1: Identify the forces acting on the system
The forces acting on the system include:
- The weight of the metre stick (W1)
- The weight of the small object (W2)
- The tensions in the two strings (T1 and T2)

### Step 2: Calculate the weights
1. The mass of the metre stick (m1) is 200 g, which is 0.2 kg.
- Weight of the metre stick (W1) = m1 * g = 0.2 kg * 9.81 m/s² = 1.962 N
2. The mass of the small object (m2) is 20 g, which is 0.02 kg.
- Weight of the small object (W2) = m2 * g = 0.02 kg * 9.81 m/s² = 0.1962 N

### Step 3: Set up the equations of equilibrium
Since the system is in equilibrium, the sum of the vertical forces must equal zero, and the sum of the moments about any point must also equal zero.

1. **Vertical Forces:**
\[ T1 + T2 = W1 + W2 \]
\[ T1 + T2 = 1.962 N + 0.1962 N = 2.1582 N \]

2. **Moments about the left end of the stick:**
Taking moments about the left end (point A), we have:
- The moment due to T2 (acting at 1 m from the left end) is counterclockwise.
- The moment due to W1 (acting at the center of the stick, 0.5 m from the left end) is clockwise.
- The moment due to W2 (acting at 0.7 m from the left end) is clockwise.

The moment equation is:
\[ T2 \cdot 1 = W1 \cdot 0.5 + W2 \cdot 0.7 \]
Substituting the values:
\[ T2 \cdot 1 = (1.962 N \cdot 0.5) + (0.1962 N \cdot 0.7) \]
\[ T2 = 0.981 N + 0.13734 N = 1.11834 N \]

### Step 4: Solve for T1
Using the vertical forces equation:
\[ T1 + T2 = 2.1582 N \]
Substituting T2:
\[ T1 + 1.11834 N = 2.1582 N \]
\[ T1 = 2.1582 N - 1.11834 N = 1.03986 N \]

### Final Results
- Tension in the first string (T1) = 1.03986 N
- Tension in the second string (T2) = 1.11834 N