A steel ring of radius r and cross-sectional area A is fitted onto a wooden disc of radius R `(R gt r).` If the Young's modulus of steel is Y, then the force with which the steel ring is expanded is ..........

Options

A. `AY (R )/(r )`
B. `AY [ (R-r)/(r )]`
C. `Y/A [ (R-r)/(r )]`
D. `(Yr)/(AR)`

Answer

B

Explanation

Circumference of wooden disc `=2pi R`
Circumference of steel ring `L = 2pi r `
Increase in circumference of steel ring
`l= 2pi R - 2pi r`
`= 2pi (R-r)`
Strain `= l/L = (2pi (R-r))/( 2pi r ) = (R-r)/(r)`
Now, Young.s modulus `Y= ("stress")/("strain") = ((F)/(A))/( (R-r)/(r))`
`therefore F = AY [ (R-r)/(r)]`