A balance is made of a light rigid rod free to rotate about a point not at the centre of the rod. When a known mass x is placed in the left pan, it is balanced by a mass y placed in the right pan. Similarly when the mass x is placed in the right pan, it is balanced by a mass z in the left pan. Neglecting the masses of the pans, x is 

Since the balance is not symmetric, the distances from the pivot to the left and right pans are different. Let’s define:

• Distance from pivot to left pan = l
• Distance from pivot to right pan = r

First Situation (x in left pan balanced by y in right pan):

For balance:

x × l = y × r

So,

x = (y × r) / l (Equation 1)

Second Situation (x in right pan balanced by z in left pan):

x × r = z × l

So,

x = (z × l) / r (Equation 2)

Equating both values of x:

(y × r) / l = (z × l) / r

Cross multiplying:

y × r² = z × l²

Or,

y / z = l² / r²

Final Expression for x:

From either Equation 1 or 2, combining both:

x = √(y × z)

Final Answer:

x = √(y × z)