A block of mass `m` is connected to a spring (spring constant `k`). Initially the block is at rest and the spring is in its natural length. Now the system is released in gravitational field and a variable force `F` is applied on the upper end of the spring such that the downward acceleration of the block is given as `a=g-alphat`, where `t` is time elapses and `alpha=1m//s^(2)`, the velocity of the point of application of the force is:

Options

A. `(malpha)/k-"gt"-(alphat^(2))/2`
B. `(malpha)/k-"gt"-alphat^(2)`
C. `(malpha)/k-"gt"+(alphat^(2))/2`
D. `(malpha)/k+"gt"+(alphat^(2))/2`

Answer

C

Explanation

Let thhe elongation of the spring be `x`. From the N.L.M of the block
`mg-kx=m(g-t)`
`kx=mt`
or `k(I-I_(0))=mt`
Differentiatinng w.r.t time
`k((dl)/(dt))=m`
`V=m/k-V^(')=m/k-int_(0)^(1)(g-t)dt`
`=m/k-"g"t+(t^(2))/2`