A bucket of water is hung from a spring balance. A solid piece of iron is suspended in the water using a thread so that it is fully immersed but does not touch the sides or the bottom of the bucket. How does the reading of the spring balance change when the iron piece is immersed in the water?

Introduction / Context:
This question illustrates the application of Archimedes principle and Newton's third law in a slightly indirect situation. A bucket of water is suspended from a spring balance, and then an iron piece is immersed in the water without touching the bucket itself. Many students are unsure whether the reading on the spring balance will increase, decrease, or remain the same. Understanding this scenario requires careful analysis of forces, buoyancy, and the interaction between the iron piece, water, and bucket.

Given Data / Assumptions:

• A bucket filled with water is hung from a spring balance.
• The initial reading shows the weight of the bucket plus water.
• An iron piece is suspended by a separate thread and fully immersed in the water.
• The iron piece does not touch the sides or the bottom of the bucket.

Concept / Approach:
When the iron piece is immersed in water, it experiences an upward buoyant force equal to the weight of the displaced water, according to Archimedes principle. For this buoyant force to exist, the water must exert an upward force on the iron piece. By Newton's third law, the iron piece exerts an equal and opposite downward force on the water. This additional downward force is transmitted to the bucket and then to the spring balance. Therefore, the total downward force acting on the spring balance increases by an amount equal to the buoyant force, causing the reading to increase. The depth of immersion does not change the magnitude of buoyant force as long as the volume of displaced water remains the same.

Step-by-Step Solution:
Step 1: Initially, the spring balance reading equals the weight of the bucket plus the water.Step 2: When the iron piece is submerged, it displaces a certain volume of water and experiences an upward buoyant force F_b.Step 3: The water exerts this upward force on the iron piece; by Newton's third law, the iron piece exerts a downward force of equal magnitude F_b on the water.Step 4: This downward force on the water is transmitted to the bucket and then to the spring balance, effectively increasing the total load.Step 5: Therefore, the spring balance reading becomes the original weight plus the additional downward force F_b, so the reading increases.

Verification / Alternative check:
Imagine replacing the immersed iron piece and thread with an equivalent extra volume of water that exactly equals the displaced water. The system would then simply have more water in the bucket, clearly increasing its weight and, thus, the spring balance reading. Since the effect of the immersed body is to make the water support part of its weight via buoyancy, the net effect on the bucket is the same as adding that much weight of water. This analogy confirms that the reading must increase when the iron piece is immersed.

Why Other Options Are Wrong:
Option B suggests that the reading decreases, which would require a reduction in the total downward force, contradicting the fact that the water now has to support part of the iron's weight. Option C states no change, which would mean the downward reaction from the water on the iron is ignored, violating Newton's third law. Option D claims that the reading varies with depth, but for a fully immersed object of fixed volume, the buoyant force is constant and independent of depth, so the additional load is fixed. Only option A stating that the reading increases is consistent with fluid mechanics and dynamics.

Common Pitfalls:
One common mistake is to think only about the weight of the iron being supported by the thread and to ignore the action reaction pair between the water and the iron piece. Another pitfall is to assume that buoyant force somehow reduces the total load on the spring balance, when in fact it represents an extra downward force transmitted through the water to the bucket. To avoid confusion, always draw free body diagrams for each object separately and carefully apply Newton's third law and Archimedes principle.

Final Answer:
The reading of the spring balance increases