A point particle of mass m, moves long the uniformly rough track PQR as shown in figure. The coefficient of friction, between the particle and the rough track equals `mu`. The particle is released, from rest from the point P and it comes to rest at a point R. The energies, lost by the ball, over the parts, PQ and QR, of the track, are equal to each other, and no energy is lost when particle changes direction from PQ to QR.
The value of the coefficient of friction `mu` and the distance x `(=QR)`, are, respectively close to:

Options

A. `0.2` and `3.5m`
B. `0.29` and `3.5m`
C. `0.29` and `6.5m`
D. `0.2` and `6.5m`

Answer

B

Explanation

Energy lost over path PQ
`=mu mg cos30^(@)xx4`........(i)
Energy lost over path QR
`=mu mg x`...........(ii)
By (i) and (ii)
`mu mg x=mu mg cos 30^(@)xx4`
`implies x=4xx(sqrt(3))/(2)=2sqrt(3)=3.45m~~3.5m`

Since , From Q to R energy lost is half of the total energy
`mu mg x =(1)/(2)mg himplies mu =0.29`.
Aliter : Given, `(mu mg h)/(tan theta)=mgh-(mu mgh)/(tan theta)`
`(2mu)/(tan theta)=1impliesmu=(tantheta)/(2)=(tan30^(@))/(2)impliesmu=0.29`
`x=(h)/(tan theta)=2sqrt(3)~~3.5m`