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The axle of a pulley of mass 1 kg is att...

The axle of a pulley of mass `1 kg` is attached to the end of an elastic string of length `1m`, cross- sectional area `10^(-3) m^(2)` and young's modulus `2xx10^(5) Nm^(-2) `,whose other end is fixed to the ceiling. A rope of negligible mass is placed on the pulley such that its left end is fixed to the ground and its right end is hanging freely , from the pulley which is at rest in equilibrium . Now, the rope and the pulley can be neglected. (Given, `g = 10 ms^(-2)`)

The elongation of the string before applying force is

A

`35 cm `

B

`30 cm`

C

`0.5 cm`

D

`0.05 cm`

Text Solution

Verified by Experts

The correct Answer is:
C
Force constant of string is
`k =("Force")/("Elongation")`
` = F/(FL//Y)`
or ` k =( YA)/L = (1xx10^(5)xx10^(-3))/1`
`= 200 Nm^(-1)`
Initial elastic potential energy of string
`= 1/2k(0.05)^(2)`
` 1/2 (200)(25xx10^(-4))`
`= 25xx10^(-2) J`
Let after force `F = 10 N` is applied extra enlongation is `x`, then
`0.25+30x = 1/2(200)(x^(2)+25xx10^(-4)+0.1x)`
`0.25+30x = 100x^(2)+0.25 + 10x`
`:. 100x^(2) = 20x`
rArr `x = 20/100 = 1/5 m = 0.2 m `
`= 20 cm `
`x^(max) = 20+5 = 25 cm `
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