Atwood machine mass at infinity help?

Question

Finally, suppose m1 -> infinity, while m2 remains finite. What value does the the magnitude of the tension approach?

I think the answer is infinity but i'm not sure.

Answer 1

The answer is not infinity.
Consider FBDs of each mass with the direction of motion of m1 as positive

m1*g-T=m1*a

T-m2*g=m2*a

assuming mass less cord and mass less, friction less pulley
the accelerations are equal
a=(T-m2*g)/m2

m1*g-T=m1*(T-m2*g)/m2
do some algebra
m1*g-T=m1*T/m2-m1*g
2*m1*g=T*(1+m1/m2)
2*m1*m2*g=T*(m2+m1)
2*m1*m2*g/(m2+m1)=T
now take the limit of T as m1->infinity
T=2*m2*g

this is intuitively correct since the maximum acceleration of m1 is -g, the cord transfers the acceleration to m2, which is being acted on by gravity downward and an upward acceleration of g. Therefore the maximum acceleration of m1 is 2*g upward.
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