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Question:

A block of mass 2M is attached to a massless spring with spring constant k. This block is connected to two other blocks of masses M and 2M using two massless pulleys and strings. The accelerations of the blocks are a1, a2, and a3 as shown in the figure. The system is released from rest with the spring in its unstretched state. The maximum extension of the spring is X0. Which of the following option(s) is/are correct?

When spring achieves an extension of x0/2 for the first time, the speed of the block connected to the spring is √(3gM/5k)

a2 - a1 = a1 - a3

At an extension of x0/4 of the spring, the magnitude of acceleration of the block connected to the spring is 3g/10

x0 = 4Mg/k

Solution:

Correct option is D.
a2 - a1 = a1 - a3
Ref. image
2a1 = a2 + a3
a1 - a3 = a2 - a1
for other options use m equivalent
Ref. image
T'g = 2(2m)(m)/(2m+m) = 4m/3
2T'g = 8m/3
meq. = 4m(2m)/(m+2m) = 8m/3
Ref. Image
(1/2)kx0^2 = (8mg/3)x0
x0 = 16mg/3k
Vx0/2 = vmax = x0/2ω = x0/2√(k/(2m + 8m/3)) = x0/2√(3k/14m) = √(3g/2)
(a)x0/4 = x0/4ω^2 = x0/4(3k/14m) = 3kx0/4(2m) = 8g/21