A uniform meter scale balances at the 40 cm mark when weights of 10g and 20g are suspended from the 10cm and 20cm marks. The weight of the metre scale is

Options

A. 50 g
B. 60 g
C. 70 g
D. 80 g

Answer

To solve the problem, we will use the concept of torque and balance of moments. Here’s a step-by-step solution:

### Step 1: Understand the Setup
We have a uniform meter scale that balances at the 40 cm mark. We are given two weights: 10 g suspended at the 10 cm mark and 20 g suspended at the 20 cm mark. We need to find the weight of the meter scale.

### Step 2: Identify the Forces and Their Distances
- The weight of the scale (W) acts at its center of mass, which is at the 50 cm mark (since it is uniform).
- The 10 g weight acts at the 10 cm mark.
- The 20 g weight acts at the 20 cm mark.

### Step 3: Calculate the Distances from the Balance Point
The balance point is at the 40 cm mark. We need to find the distances from this point to where the weights are applied:
- Distance from 40 cm to 10 cm (for 10 g): \( 40 - 10 = 30 \) cm (clockwise)
- Distance from 40 cm to 20 cm (for 20 g): \( 40 - 20 = 20 \) cm (clockwise)
- Distance from 40 cm to 50 cm (for W): \( 50 - 40 = 10 \) cm (anticlockwise)

### Step 4: Set Up the Torque Equation
For the system to be in balance, the total clockwise torque must equal the total anticlockwise torque.

The torque due to the weights is:
- For the 10 g weight: \( 10 \, \text{g} \times 30 \, \text{cm} \)
- For the 20 g weight: \( 20 \, \text{g} \times 20 \, \text{cm} \)

The torque due to the weight of the meter scale (W) is:
- For the scale: \( W \times 10 \, \text{cm} \)

Setting up the equation:
\[ 10 \, \text{g} \times 30 \, \text{cm} + 20 \, \text{g} \times 20 \, \text{cm} = W \times 10 \, \text{cm} \]

### Step 5: Substitute and Simplify
Calculating the left side:
\[ 10 \times 30 = 300 \, \text{g cm} \]
\[ 20 \times 20 = 400 \, \text{g cm} \]
Adding these gives:
\[ 300 + 400 = 700 \, \text{g cm} \]

Now, substituting back into the torque equation:
\[ 700 = W \times 10 \]

### Step 6: Solve for W
To find W, divide both sides by 10:
\[ W = \frac{700}{10} = 70 \, \text{g} \]

### Final Answer
The weight of the meter scale is **70 g**.

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