Question

A false balance has equal arms. An object weighs ${w_1}$ when placed in one pan and ${w_2}$ when placed in another pan. Then the weight $w$ of the object is:
A. $\sqrt {{w_1}{w_2}}$
B. $\dfrac{{{w_1} + {w_2}}}{2}$
C. $\dfrac{{{w_1}^2 + {w_2}^2}}{2} - 1$
D. $\sqrt {{w_1}^2 + {w_2}^2}$

Answer 1

The gravitational force is proportional to the mass of the body, the more mass the more gravitational pull would act on the body. That is why the pan with heavyweight would be below and the pan with less weight would be above. If the weights are equal then both pans were in equilibrium or balanced. When two bodies are balanced (in equilibrium) the sum of the clockwise moments is equal to the sum of the anticlockwise moments.

Formula used:
$F = mg$
Where $F$ is the gravitational force, $m$ is the mass of the object, $g$ is the acceleration due to gravity.

Complete step by step solution:
Let’s suppose that the mass of pan 1 and pan 2 are $x$ and $y$ respectively,
Force on pan 1, $F1 = ({w_1} + x)g$
Force on pan 2, $F2 = ({w_2} + y)g$
For pan 1 $({w_2} + x)g = (w + y)g$ ……………… (1)
For pan 2 $({w_1} + y)g = (w + x)g$ …………… (2)
Subtracting equation (2) from (1)
$w - {w_2} = {w_1} - w$
$\Rightarrow w = \dfrac{{{w_1} + {w_2}}}{2}$
So the correct option is B.

Note:
In nature we can say an object is in balance if it is not moving, So when an object is moving it is in the state of equilibrium.
The center of gravity is the average position of the force of gravity on a body. Sometimes it is at the object’s geometric center for example ruler, where other times it isn’t (e.g. ruler with an eraser on one end). An object can be balanced or in equilibrium if it is supported directly under its center of gravity.
Common balance can be defined as the balance which is having each arm suspended. The unidentified mass is kept in one arm and the known mass in another until they both become equivalent. Therefore this balance is working on the principle of the moment of weights.