A wire of length L is hanging from a fixed support. The length changes to `L_(1) and L_(2)` when masses `M_(1)and M_(2)` are suspended respectively from its free end. Then L is equal to

Options

A. `(L_(1) + L_(2))/(2)`
B. `sqrt(L_(1)L_(2))`
C. `(L_(1) M_(2) + L_(2) M_(1))/(M_(1) + M_(2))`
D. `(L_(1) M_(2) - L_(2) M_(1))/(M_(2) - M_(1))`

Answer

D

Explanation

Let L be the original length of the wire

When the mass `M_(1)` is suspended from the wire, change in length of wire is `Delta L_(1) = L_(1) - L`
When the mass `M_(2)` is suspended from it, change in length of wire is
`Delta L_(2) = L_(2) - L`
From figure (b), `T_(1) = M_(1)g` ...(i)
From figure (c ), `T_(2) = M_(2) g` ...(ii)
As Young.s modulus, `Y= (T_(1) L)/(A Delta L_(1)) = (T_(2) L)/(A Delta L_(2))`
`:. (T_(1))/(Delta L_(1)) = (T_(2))/(Delta L_(2)) or (T_(1))/(L_(1) - L) = (T_(2))/(L_(2) - L)`
or `(M_(1)g)/(L_(1) - L) = (M_(2)g)/(L_(2) - L)` (Using (i) and (ii))
or `M_(1) (L_(2)- L) = M_(2) (L_(1) - L) or M_(1) L_(2) - M_(1) L = M_(2) L_(1) - M_(2)L`
or `L(M_(2) - M_(1)) = L_(1) M_(2) - L_(2) M_(1)`
or `L= (L_(1) M_(2) - L_(2) M_(1))/(M_(2) - M_(1))`