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JEE Advanced 1985 · Mechanical Properties of Fluids

The spring balance AA reads 2 kg2\,\text{kg} with a block mm suspended from it. AA balance BB reads 5 kg5\,\text{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 the beaker as shown in the figure. In this situation:

Answers: B. the balance BB will read more than 5 kg5\,\text{kg} · C. the balance AA will read less than 2 kg2\,\text{kg} and BB will read more than 5 kg5\,\text{kg}

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Written solution

Step 1: Analyze the initial state in air.

When the block is suspended in air, balance AA measures the tension in the string, which equals the weight of the block.

W=2 kgfW = 2\,\text{kgf}

Balance BB measures the weight of the beaker and the liquid.

Wliq=5 kgfW_{\text{liq}} = 5\,\text{kgf}

Step 2: Analyze the forces on the immersed block.

When the block is lowered into the liquid, the liquid exerts an upward buoyant force FBF_B on the block.

T+FB=WT + F_B = W T=W−FBT = W - F_B T=2−FB<2 kgfT = 2 - F_B < 2\,\text{kgf}

Since balance AA reads the tension TT, its reading becomes less than 2 kg2\,\text{kg}.

Step 3: Analyze the forces on the beaker and liquid.

By Newton's third law, the block exerts an equal and opposite downward force FBF_B on the liquid.

N=Wliq+FBN = W_{\text{liq}} + F_B N=5+FB>5 kgfN = 5 + F_B > 5\,\text{kgf}

Since balance BB reads the normal force NN, its reading becomes greater than 5 kg5\,\text{kg}.

Step 4: Verify using the total system equilibrium.

For the combined system of the block, beaker, and liquid, the total upward force must balance the total downward weight.

T+N=W+WliqT + N = W + W_{\text{liq}} T+N=2+5=7 kgfT + N = 2 + 5 = 7\,\text{kgf}

The sum of the readings of AA and BB remains constant at 7 kg7\,\text{kg} regardless of the immersion.

Answer: The reading of balance AA is less than 2 kg2\,\text{kg} and the reading of balance BB is greater than 5 kg5\,\text{kg}.

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