Two blocks of masses m1=1.7 kg and m2=3.7 kg are put on a frictionless horizontal surface. A spring of stiffness k=970 N/m and negligible mass is placed between the blocks and they are pushed together to compress the spring by distance d from its equilibrium length. The configuration is fixed by a cord that holds the blocks. When the cord is burned, the first block moves to the left with a speed of v1=1.9 m/s. Calculate the initial compression of the spring, d.
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Expert's answer
2021-07-19T09:50:45-0400
m1=1.7kg;m2=3.7kg;k=970mN
Spring is compressed:
v1′=0;v2′=0
Δx′=d
E=Ek′+Ep′
Ek′=2m1v1′2+2m2v2′2=0
Ep′=2kΔx′2=2kd2
E=2kd2
Spring is released:
v1=1.9sm;v2−?
Δx=0
E=Ek+Ep
Ep=2kΔx2=0
Ek=2m1v12+2m2v22
by the law of conservation of momentum
m1v1+m2v2=m1v1′+m2v2′
m1v1+m2v2=0
v2=−m2m1v1=−3.71.7∗1.9=−0.87
Ek=21.7∗1.92+23.7∗(−0.87)2=4.47J
E=Ep+Ek=4.47
E=2kd2
d=k2E=9702∗4.47≈0.07m
Answer: initial compression of the spring, d = 0.07m
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