The circular vertical section of the fixed track shown is smooth with radius `r=0.9cm` and the horizontal straight section is rough with `mu=0.1` . A block of mass `1kg` is placed at point `'Q'` and given a horizontal velocity of `sqrt(3)m//s` towards the spring. Distance `QS=40cm` and maximum compression in the spring is `10cm` during the motion `(g=10m//s^(2)):`

Options

A. The force constant of the spring is `200N//m`
B. The velcoity with which block returns to point `'Q'` is `1m//s`
C. At point `P` its velocity will be `0.8m//s`
D. At point `P`, the normal reaction on the block is less than `55N`

Answer

A, B, C

Explanation

`Q rarr S`
`- mu mg xxS=0=0+(1)/(2)k(0.1)^(2)-(1)/(2)xx1xx(sqrt(3))^(2)`
`-0.1xx1xx10xx0.5=(k)/(200)-(3)/(2)`
`k=200N//m`
`Q rarr S rarr Q`
`-mu mg xx0.5xx2=(1)/(2)mv_(Q)^(2)-(1)/(2)m u^(2)`
`-1=(1)/(2) xx1xxv_(Q)^(2)-(1)/(2)xx1xx3`
`v_(Q)=1m//s`
`v_(P)^(2)=v_(Q)^(2)-2(g)(2r)`
`v_(P)=sqrt(1^(2)-4xx10xx(0.9)/(100))=(8)/(10)=0.8m//s`
`N_(P)=(mv_(P)^(2))/(r)-mg=(1xx64xx100)/(100xx0.9)-10=(55)/(0.9)N`.