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Net Force

A spring bala...

Question

A spring balance reads $W_{1}$ , when a ball is suspended from it. A weighing machine reads $W_{2}$ , when a tank of liquid is kept on it. When the ball is immersed in the liquid, the spring balance reads $W_{3}$ and the weighing machine reads $W_{4}$ . Which of the following relationship is correct.

W_{1} + W_{2} = W_{3} + W_{4}

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W_{1} + W_{3} = W_{2} + W_{4}

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W_{1} + W_{4} = W_{2} + W_{3}

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W_{1} + W_{2} + W_{3} = W_{4}

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Solution

The correct option is A $W_{1} + W_{2} = W_{3} + W_{4}$
Let mass of ball be $m$ , mass of tank be $M$ , Buoyancy force by liquid in tank on the ball be $B$ , $k$ be spring constant, $x_{1}$ be elongation of spring when ball is suspended in air, and $x_{2}$ be elongation of spring when ball is suspended immersed in liquid.
$W_{1} = k x_{1} = m g$
$W_{2} = M g$ is normal force by weighing machine.
$W_{3} = k x_{2} = m g - B$
$W_{4} = m g - k x_{2} + M g$ is normal force by weighing machine.
So, $W_{1} + W_{2} = W_{3} + W_{4} = m g + M g$

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