Question

(a) The right block collides with left block and sticks to it. Find amplitude of the resulting SHM.

(b) The block starts from left wall and moves with constant speed v. All collisions are elastic. Find time period of periodic motion.

Answer 1

(a)
By momentum conservation
`2mv=(m+2m)VimpliesV=(2v)/(3)`
K.E. of combined block
`=1/2(2m+m)V^2=(3m)/(2)((2v)/(3))^2=2/3mv^2`
Let A be the amplitude
`2/3mv^2=1/2kA^2impliesA=vsqrt((4m)/(3k))`
(b) The block reaches to spring with speed v. The block compresses the spring by distance x such that `1/2mv^2=1/2kx^2`. Now block moves toward left, when it leaves contact with spring, its speed is v. The time for which block is in contact with spring `t_1=T/2=pisqrt(m/k)`.
Time taken by block to move from left wall to spring and then spring to left wall
`t_2=L/v+L/v=(2L)/(v)`
Thus block undergoes periodic motion with time period
`t_1+t_2=pisqrt(m/k)+(2L)/(v)`.