A block of mass 1 kg kept on a rough horizontal surface `(mu=0.4)` is attached to a light spring (force constant `=200`N//m) whose other end is attached to a verticle wall. The block is pushed to compress the spring by a distance d and released. Find the value(s) of 'd' for which (spring + block) system loses its entire mechanical energy in form of heat.

Options

A. 4 cm
B. 6 cm
C. 8 cm
D. 10 cm

Answer

A, C

Explanation

The system will lose its entire mechanical energy if the block finally comes to rest at the mean position. Now suppose the block comes to rest for the first time after the start at a distance `d_(1)` from the mean position on the other side, then `1/2k(d^(2)-d_(1)^(2)) -mumg(d+d_(1))=0`

(From work energy theorem, `DeltaK=0`) Rightarrow `d-d_(1)=(2mumg)/k Rightarrow d_(1)=d-(2mumg)/k`
Now, similarly if `d_(2),d_(3)` e.t.c. are the successive distances of the block from the mean position where it come to rest for the second, third times etc. then `d_(2)-d_(3)=d_(1)`
`-d_(2)=d-d_(1)=(2mumg)/k Rightarrow d_(2)=d_(1)-(2mumg)/k=d-(4mumg)/k`
`d_(3)=d_(2)-(2mumg)/k=d-(6mumg)/k`
`d_(n)=d-(2mumg)/k`
we want `d_(n)=0`
Rightarrow `d=n((2mumg)/k)`
i.e.d should be an integral multiple of `(2mumg)/k`