How to find the equation for the maximum velocity possible

[brenna_s]

Homework Statement

When an object is moving in a vertical circle attached to a string, it can withhold a maximum tension at the bottom of it's path. Derive an expression for the maximum velocity the object can sustain without the string breaking. No data is given, and everything should be variables.

Homework Equations

V minimum equals the square root of gravity x radius
Fnet equals Fg - Tension at the bottom of a vertical path
Fnet equals mass x centripetal acceleration

The Attempt at a Solution

I attempted to use Fnet equals mass x centripetal acceleration - mg but that didn't work. I solved for v min in the previous problem, and found that it was the square root of gravity x radius, but I'm not sure how to go from that to v maximum.

 

[berkeman]

Homework Statement

When an object is moving in a vertical circle attached to a string, it can withhold a maximum tension at the bottom of it's path. Derive an expression for the maximum velocity the object can sustain without the string breaking. No data is given, and everything should be variables.

Homework Equations

V minimum equals the square root of gravity x radius
Fnet equals Fg - Tension at the bottom of a vertical path
Fnet equals mass x centripetal acceleration

The Attempt at a Solution

I attempted to use Fnet equals mass x centripetal acceleration - mg but that didn't work. I solved for v min in the previous problem, and found that it was the square root of gravity x radius, but I'm not sure how to go from that to v maximum.

Welcome to the PF.

There may be more to the problem, but as stated it seems pretty simple. It seems to only involve what happens at the bottom of the circle where the maximum velocity and the maximum string tension occur. Am I wrong about that?

You know what the centripital acceleration is as a function of tangential velocity (and therefore the force), and you know the force on the mass is due to gravity. So just add those at the bottom of the circle and equate those to the maximum tension it takes to break the string?

 

[haruspex]

Fnet equals Fg - Tension at the bottom of a vertical path
Fnet equals mass x centripetal acceleration

Each of those is correct in itself, with appropriate choices of positive direction, but as written the two are not consistent in that regard.
At the bottom, centripetal acceleration is up but gravity is down.

Fnet equals mass x centripetal acceleration - mg

That is wrong. As you wrote before, centripetal acceleration is a result of the net force. The net force is the resultant of the applied forces, tension and gravity.

 

[brenna_s]

Thank you both!