Question

body
spring
B fig shows the construction of single
acting cylinder it has single intet for
the compress and it
inlet or outlet as per Stroke to be
performed.
@ when compress and enters through
the inlet it Esetts
pressure on
the piston against the! Spring force.
© when compress ait qeneéate a force
go
greater than the spring force the
piston moves from left end to
right end of the to
fondard movement the spring gate
complete
Compressed
the​

Answer 1

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Answer 2

Let the block A shift to left by x1 and block B shift to right by x2. The centre of mass of the two block system is at rest.

Hence mx1=2ms2.

or x2=x12…..(1)

and the spring force on either block is k(x0−[x1+x2]), where x0 is the initial compression in the spring.

Let the block A shift further left by dx1

∴work donw on block by spring is

dW=(k(x0−x1−x2)dx1.....(2)

=k(x0−x1−x12)dx1

∴Net work done

∫dW=∫x0/3x1=0k(x0−32x1)dx1=kx204

Ans.12kx20

Alternative Solution

Let the speeds of blocks A and B at the instant compression is x02 be vA andvB as shown in figure

[l0] natural length of spring ]

No external forces act on the system in the horizontal direction

Applying conservation of momentum in horizontal direction

Initial momentum = final momentum

0=m(−vA)+2mvB

or vA=2vB......(1)

from conservation of energy

12kx20=12k(x02)2+12mv2A+122mv2B....(2)

from (1) and (2) we get

12mv2A=14kx20

work done on block A by spring = change in kinetic energy of block A

=12mv2A=14kx20