Suppose that in the same Atwood...?

Question

Suppose that in the same Atwood setup another string is attached to the bottom of m1 and a constant force f is applied,retarding the upward motion of m1. If m1=6.4 kg and m2=8.21 kg, what value of f will reduce the acceleration of the system by 50%?

Answer 1

I am assuming the Atwood machine is ideal such that there is no mass and no friction in the pulley
FBD for m1:
T1-6.4*9.81=6.4*a
and
T2-6.4*9.81-f=6.4*a/2

for m2
8.21*9.81-T1=8.21*a

8.21*9.81-T2=8.21*a/2

solve for f
a=1.215 m/s^2
T1=70.56
T2=75.55
f=16.66 N

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