In a physical balance working on the principle of moments, when 5 mg weight is placed on the left pan, the beam becomes horizontal. Both the empty pans of the balance are of equal mass. Which of the following statements is correct ?
In a physical balance working on the principle of moments, when 5 mg weight is placed on the left pan, the beam becomes horizontal. Both the empty pans of the balance are of equal mass. Which of the following statements is correct ?
A
Every object that is weighted using this balance appears lighter than its actual weight
B
Left arm is shorter than the right arm
C
Both the arms are of same length
D
Left arm is longer than the right arm
Text Solution
AI Generated Solution
The correct Answer is:
To solve the problem, we need to analyze the situation described in the question using the principle of moments (or torque). Here's a step-by-step solution:
### Step 1: Understand the Setup
We have a physical balance with two pans. When a 5 mg weight is placed on the left pan, the beam becomes horizontal. This indicates that the torques acting on both sides of the balance are equal.
### Step 2: Define the Forces
Let:
- \( m \) be the mass of the weight on the left pan (5 mg = 0.005 g).
- \( M \) be the mass of each empty pan (equal mass).
- \( g \) be the acceleration due to gravity.
The total force on the left pan is:
\[ F_{left} = mg + M \cdot g \]
The total force on the right pan (which is empty) is:
\[ F_{right} = M \cdot g \]
### Step 3: Set Up the Torque Equation
The torque (\( \tau \)) is given by the product of the force and the distance from the pivot point. Let:
- \( r_1 \) be the distance from the pivot to the left pan.
- \( r_2 \) be the distance from the pivot to the right pan.
For the balance to be horizontal, the torques must be equal:
\[ F_{left} \cdot r_1 = F_{right} \cdot r_2 \]
Substituting the forces:
\[ (mg + Mg) \cdot r_1 = Mg \cdot r_2 \]
### Step 4: Simplify the Equation
Since \( g \) is common in all terms, we can cancel it out:
\[ (m + M) \cdot r_1 = M \cdot r_2 \]
### Step 5: Analyze the Equation
Rearranging gives:
\[ r_2 = \frac{(m + M)}{M} \cdot r_1 \]
Since \( m = 0.005 \, g \) and \( M \) is the mass of the empty pan, we can see that \( r_2 \) is greater than \( r_1 \) because \( m + M > M \).
### Step 6: Conclusion
Since the distance \( r_2 \) (right pan) is greater than \( r_1 \) (left pan), the left arm is shorter than the right arm. Thus, the correct statement is that the left arm is shorter than the right arm.
### Final Answer
The correct statement is: **The left arm is shorter than the right arm.**
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