Circular Motion Problem: Multiple Masses

[theed21]

Homework Statement

Three masses are connected by strings and swung in a vertical circle. You are holding string 1 which is attached to a mass attached to string 2 likewise attached to another mass string 3. Which string has the greatest tension? Which string has the smallest tension?

Homework Equations

Centripetal force = m(v^2)/r

The Attempt at a Solution

I said that String 3 has the greatest tension because the object will make a bigger circle because it is farther away from the center. And the tension force is the same as centripetal force in this case which is mv^2/r. velocity has r in the numerator so when you square it, r ends up on top in the force equation. So as the bigger the radius, the larger tension force. My friend insists that all three tension string forces are equal.

 

[collinsmark]

Hello theed21,

Welcome to Physics Forums!

Homework Statement

Three masses are connected by strings and swung in a vertical circle. You are holding string 1 which is attached to a mass attached to string 2 likewise attached to another mass string 3. Which string has the greatest tension? Which string has the smallest tension?

By that I am assuming that that string/mass system is swinging around "you." In other words I'm assuming that your hand is at the center of rotation.

Homework Equations

Centripetal force = m(v^2)/r

The Attempt at a Solution

I said that String 3 has the greatest tension because the object will make a bigger circle because it is farther away from the center. And the tension force is the same as centripetal force in this case which is mv^2/r. velocity has r in the numerator so when you square it, r ends up on top in the force equation. So as the bigger the radius, the larger tension force. My friend insists that all three tension string forces are equal.

Both you and your friend should give this problem a little more thought. Drawing a free-body diagram might help as well, if you wish to explain the results mathematically.

But let me at least offer this for consideration. Take a look at one of the masses in isolation. For example, let's take M₂, which has a string 2 on one side and string 3 on the other. In other words, ignoring gravity* there are two forces acting on M₂: T₃ and T₂.

If T₃ ≥ T₂, as you and/or your friend proposed, then how is it that M₂ is accelerating in the direction of T₂ to maintain the circular motion?

*(I'm not sure if gravity is supposed to fit into this problem. The problem statement did indicate the the masses are swung in a vertical circle, which would normally indicate that gravity is another force to consider. Are you sure you don't mean horizontal circle? I suppose it doesn't matter though. The mathematics would be more difficult when considering gravity but the qualitative answer wouldn't change, assuming all the strings remain on a straight line.)