Problem 51 - Master Resource Book in JEE Main Physics

Question

In measurement of mass of a given object by the principle of moments, the meter scale is hung from its mid-point. A known weight of mass \(2 \mathrm{~kg}\) is hung at one end of meter scale and unknown weight of mass \(\mathrm{m} \mathrm{kg}\) is hung at \(20 \mathrm{~cm}\) from the centre on other side. The value of \(m\) is (a) \(2 \mathrm{~kg}\) (b) \(5 \mathrm{~kg}\) (c) \(2.5 \mathrm{~kg}\) (d) \(0.8 \mathrm{~kg}\)

Step-by-Step Solution

Answer

The value of \(m\) is \(5 \mathrm{~kg}\).

Step 1: Understanding the Principle of Moments

The principle of moments states that for a system in equilibrium, the total clockwise moments about a pivot point must equal the total counter-clockwise moments about that same point. In this context, the pivot point is the midpoint of the meter scale.

Step 2: Identifying Forces and Distances

Identify the forces and the distances from the pivot point. The known weight of 2 kg is hung at one end of the meter scale, i.e., 50 cm from the midpoint since the total length of the meter scale is 100 cm. The unknown weight m kg is hung 20 cm from the midpoint.

Step 3: Setting Up the Moment Equations

Set up the equation for the balance of moments. The moment due to the known weight (2 kg) is calculated using the formula: Moment = Force × Distance. Thus, the moment is \(2 \times 50\). The moment due to the unknown weight m is \(m \times 20\).

Step 4: Solving the Moment Equation

Since the system is in equilibrium, set the clockwise moments equal to the counter-clockwise moments: \(2 \times 50 = m \times 20\). Simplify the equation to find the value of m: \(100 = 20m\).

Step 5: Calculating the Value of m

Solve the equation \(100 = 20m\) to find \(m\). Divide both sides by 20 to get \(m = \frac{100}{20} = 5\).

Key Concepts

Equilibrium

Equilibrium refers to a state where all forces or moments acting on a system are perfectly balanced, leaving the system in a stable condition with no net movement. In the scenario of a meter scale used to measure mass, achieving equilibrium is crucial for accurate mass determination.

When a system is in equilibrium, it means that the sum of clockwise moments around a pivot point is equal to the sum of counter-clockwise moments around the same pivot. This balance of forces ensures that the meter scale remains horizontal and stable.

• The meter scale behaves like a seesaw, with the pivot point functioning as the fulcrum.
• Forces acting on either side must be carefully balanced to maintain stability.
• If one side exerts more torque or moment than the other, the scale will tip, indicating a state of imbalance.

Understanding equilibrium allows for precise applications of the principle of moments, ensuring the system remains steady and accurate measurements are achieved.

Moment Equation

The moment equation is a fundamental part of mechanics used to maintain equilibrium in systems like the meter scale. A moment, also known as torque, is the measure of the force causing an object to rotate. It depends on two factors: the magnitude of the force and the distance from the pivot point.

The moment equation can be expressed as: \[ \text{Moment} = \text{Force} \times \text{Distance} \]These calculations help analyze cases where a pivot creates rotational effects. For the exercise:

• The known weight of 2 kg creates a moment of \(2 \times 50 = 100 \text{ kg cm}\).
• The unknown mass \(m\) creates a moment of \(m \times 20\text{ kg cm}\).

Using these formulas ensures that the moments calculated accurately reflect the conditions necessary for equilibrium.

By equating and solving the moments due to known and unknown weights, one achieves an understanding of the mass \(m\) required for balance, thus demonstrating the principle of moments.

Force and Distance Calculation

Force and distance are key contributors in calculating moments, essential for achieving equilibrium in systems. In the exercise provided, calculating these factors requires careful analysis and understanding of their interaction.

• **Force** is straightforward—it’s the weight force due to gravity acting on the masses hung on the meter scale.
• **Distance** refers to the perpendicular distance from the pivot (i.e., the midpoint of the meter scale) to where the force acts.

Forces result in rotational effects when applied at a distance from the pivot, termed the lever arm. In the provided setup:

• The known weight (2 kg) acts at a distance of 50 cm, producing a substantial clockwise moment.
• The unknown mass \(m\) needs to balance this with a counter-clockwise moment acting 20 cm away from the pivot.

This understanding aids in correctly positioning these forces to achieve necessary balance, highlighting their critical role in the system's equilibrium. By accurately determining force and distance, one can effectively apply the moment equation to solve for unknown quantities.