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Question

Mass of block A is m and that of block B is 2m. Spring constant is k and there is no friction. System is released from rest with the spring unstretched. Pulley is massless. Identify the correct statement(s).


  1. Maximum extension of the spring is 4mgk
  2. Acceleration of block B is g3 downwards, when extension in the spring is mgk
  3. Maximum extension in the spring is 2mgk
  4. Acceleration of block B is g3 downwards, when extension in spring is 4mgk


Solution

The correct options are
A Maximum extension of the spring is 4mgk
B Acceleration of block B is g3 downwards, when extension in the spring is mgk
mA=mmB=2m
System is released from rest, so initial velocity of both blocks is 0 m/s.
Let the maximum extension in spring be xm.


At the maximum externsion in spring, both the blocks will be at rest momentarily.
Applying mechanical energy conservation,
Loss in gravitational PE of B=Gain in spring PE
(K.E is zero initially and finally)
2mgxm=12kx2m
xm=4mgk
Let the acceleration of block B be a. Then acceleration of block A will also be a (due to string constraint).


FBD of block A:


T1T2=ma
If extension in spring is x=mgk,
T2=kx=mg
T1mg=ma ...(i)
FBD of block B:


2mgT1=2ma ...(ii)
Adding (i) & (ii):
a=g3 (downwards)
Similarly, when extension in spring is 4mgk, then:
T14mg=ma ...(i) for block A.
[T2=kx=4mg]
For block B
2mgT1=2ma ...(ii)
Adding (i) & (ii):
a=2g3
when x=4mgk, block B will move upwards with acceleration 2g3.
Only option A and B are correct.

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