Figure shows a block of mass 'm' resting on a smooth horizontal surface. It is connected to a rigid wall by a massless spring of stiffness 'k'. The spring is in its natural length. A constant horizontal force F starts acting on the block towards right. Find (i) speed of the block as it moves throught a distance x, (ii) speed when the block is in equlibrium and (iii) maximum extension produced in the speing.

Explanation

When the block has moved through 'x' its speed is v(say)

The FDB of the block in this situation is shown
(i) By work energy principle, `W=DeltaK`
`W_(F)+W_(sp)=1/2mv^(2)-0`
`impliesFx x xcos0^(@)-underset(0)overset(x)intkxdx=1/2mv^(2)`
`impliesfx-1/2kx^(2)=1/2mv^(2)`
`thereforev=sqrt((2Fx-kx^(2))/(m))`
(ii) At equlibrium, net force is zero.
`F=kxorx=F/k`
Now `v=sqrt((2Fx-kx^(2)))/(m)`
`thereforev=sqrt(((2F^(2))/(k)-(F^(2))/(k))/(m))=(F)/(sqrtmk)`
(iii) At the maximum extension, speed of the block becomes zero. `thereforev=0`
`or, sqrt((2Fx-kx^(2))/(m))=0impliesx=(2F)/(k)`