A uniform rod of mass M and length L, area of cross section A is placed on a smooth horizontal surface. Forces acting on the rod are shown in the digram

Ratio of elongation in section PQ of rod and section QR of rod is

Options

A. `19:37`
B. `21:39`
C. `23:41`
D. `17:35`

Explanation

`F=((F_(1)-F_(2))/(x_(0)))x+F_(2)=alphax+beta`
Energy density at any x
`(dU)/(dV)=1/2((alphax+beta)/A)((alphax+beta)/(AY))=1/(2A^(2)Y)(alphax+beta)^(2)`
Energy stored on small segment dx
`dU=1/(2A^(2)Y)(alpha^(2)x^(2) +beta^(2)+2alphabetax)Adx`
`U=intdU=1/(2AY) int_(0)^(x_(0))(alpha^(2)x^(2)+beta^(2)+2alphabetax)dx=1/(2AY)((alpha^(2)x_(0)^(3))/3+beta^(2)x_(0)+betax_(0)^(2))`
consider section PQ
`alpha=F//L, beta=F, x_(0) =L//2`
`U_(1)=(19 F^(2)L)/(48AY)` consider section QR
`alpha=F//L , beta=3F//2, x_(0)=L//2`
`U_(2)=(37F^(2)L)/(48AY)`