Problem 18 - Master Resource Book in JEE Main Physics

Question

The spring balance \(A\) reads \(2 \mathrm{~kg}\) with a block of mass \(m\) suspended from it. A balance \(B\) reads \(5 \mathrm{~kg}\) when a beaker with liquid is put on the pan of the balance. The two balances are now so arranged that the hanging mass is inside the liquid in a beaker as shown in figure. (a) The balance A will read more than \(2 \mathrm{~kg}\) (b) The balance \(B\) will read less than \(5 \mathrm{~kg}\) (c) The balance \(A\) will read less than \(2 \mathrm{~kg}\) and \(B\) will read more than \(5 \mathrm{~kg}\) (d) The balance \(A\) will read more than \(2 \mathrm{~kg}\) and \(B\) will read less than \(5 \mathrm{~kg}\)

Step-by-Step Solution

Answer

The balance A will read less than 2 kg, and B will read more than 5 kg.

Step 1: Analyzing Spring Balance A

When the block of mass \( m \) is submerged in the liquid, it experiences a buoyant force upward, which is equal to the weight of the liquid displaced by the block. This reduces the effective weight of the block. Therefore, the reading on the spring balance \( A \) will be less than the block's actual weight of \( 2 \text{ kg} \). Thus, the reading on balance \( A \) will be less than \( 2 \text{ kg} \).

Step 2: Analyzing Balance B

The beaker with the liquid initially reads \( 5 \text{ kg} \). When the block is submerged, the buoyant force the liquid exerts on the block acts downwards on the balance \( B \). This force is equal and opposite to the reduction in weight on balance \( A \). Thus, the balance \( B \) reads more than \( 5 \text{ kg} \) because it supports the weight of the liquid plus the buoyant force.

Step 3: Conclusion of Solution

Combining the analysis from Step 1 and Step 2, we conclude that the spring balance \( A \) will read less than \( 2 \text{ kg} \), and the balance \( B \) will read more than \( 5 \text{ kg} \). The correct option is (c).

Key Concepts

Buoyant Force

Buoyant force is a fascinating concept that you might have experienced without even realizing it. Have you ever noticed that objects seem to weigh less when they are submerged in water? This is due to the buoyant force, which acts in an upward direction.

• What is Buoyant Force?
  It is the upward force exerted by a fluid on a submerged object. This force is equal to the weight of the fluid displaced by that object.

Archimedes' Principle helps us understand this: any object, partly or fully submerged in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object. When thinking about why the block in the exercise weighs less in the liquid, it's the buoyant force at play. This force reduces the block's effective weight, leading to a different reading on a balance.

Spring Balance

A spring balance is a handy tool for measuring weight. It's like a scale, but it uses a spring to determine how heavy an object is.

• How Does a Spring Balance Work?
  The balance operates by measuring the extent of stretch in the spring, which is directly proportional to the force it experiences.
• Impact of Buoyant Force
  When you submerge an object in the liquid, the buoyant force reduces the force on the spring, lowering the weight reading.

In the exercise, as the block is submerged, the spring balance 'A' shows a weight less than the actual mass of the block because the liquid supports part of the block’s mass through buoyancy.

Effective Weight

The concept of effective weight can be a bit intriguing. It refers to the perceived weight of an object when a buoyant force is applied to it.

• What is Effective Weight?
  It's the weight that an object seems to have when submerged in a fluid and experiencing a buoyant force.
• Important Insight
  Effective weight is essentially the actual weight minus the buoyant force, since the fluid helps support the object.

In our scenario, the block's effective weight decreases inside the liquid, allowing balance 'A' to recognize less weight than when the block is outside the water, while balance 'B' reflects more weight, since it must support the added buoyant force beneath the block.