Two identical blocks A and B, each of mass `m=3kg`, are connected with the help of an ideal spring and placed on a smooth horizontal surface as shown in Fig. Another identical blocks C moving velocity `v_0=0.6(m)/(s)` collides with A and sticks to it, as a result, the motion of system takes place in some way
Based on this information answer the following questions:
Q. After the collision of C and A, the combined body and block B would

Option1
oscillate about centre of mass of system and centre of mass is at rest.
Option2
oscillate about centre of mass of system and centre of mass is moving.
Option3
oscillate but about different location other than the centre of mass.
Option4
not oscillate.

Options

A. oscillate about centre of mass of system and centre of mass is at rest.
B. oscillate about centre of mass of system and centre of mass is moving.
C. oscillate but about different location other than the centre of mass.
D. not oscillate.

Answer

B

Explanation

As `C` collides with A and sticks to its, the combined mass moves rightwards to compress the spring and hence B moves rightwards due to the spring force, i.e., B accelerates and the combined mass decelerates. The deformation in the spring is changing and the centre of mass of the system continues to move rightwards with contant speed, while both the blocks oscillate about centre of mass of the system.
The velocity of the combined mass just after collision is
`mv_0=2mv`
`v=(v_0)/(2)=0.3(m)/(s)`
Velocity of centre of mass of system.
`v_(CM)=(2mv+0)/(3m)=(2v)/(3)=(v_0)/(3)=0.2(m)/(s)`
time period with which block B and the combined mass oscillate about centre of mass could be computed by using the reduced mass concept.
`T=2pisqrt((2m)/(3k))=(pi)/(5sqrt(10))s`
Oscillation energy of the system is
`E=(2mv^2)/(2)=0.27J`
the translational kinetic energy of the centre of mass of system is
`E_(CM)=(3mv^2)/(2)=0.18J`
The remaining energy is oscillating between kinetic and potential energy during the motion of blocks. For maximum compression in spring either we can use cantre of mass approach or energy approach, here we are using the second method.
Oscillation energy `=` Maximum elastic potential energy
`0.09=(1)/(2)kx_m^2impliesx_m=3sqrt(10)mm`