Sphere Under Gravity Problem (Isaac Physics)

[bobbricks]

I'm stuck on this problem:
A small sphere is attached to a fixed point by a string of length =30cm , and whirls round in a vertical circle under the action of gravity at such speed that the tension in the string when the sphere is at its lowest point is three times the tension when the sphere is at its highest point.
From: https://isaacphysics.org/questions/sphere_under_gravity

I initially thought it would be constant velocity but that produces the wrong answer and after looking at the hints, I found the velocity at the top is different to the velocity at the bottom.

I've tried using ideas of energy where the Energy at the top=mg0.6+0.5mv² and Energy at the bottom= 0.5mv²

Also, resolving forces: At the top F=T+mg where F is the centripetal force. At the bottom F=3T-mg.

I've also tried working out the minimum velocity required using a=v²/r where a=9.81 and r=0.3m. I've tried to use Work=forcexdisplacement so F=W/d and d=0.6 and W=energy at the top or W= energy at the bottom, to produce 2 separate equations. Still stuck.

 

[Doc Al]

You'll need both energy analysis (conservation) and force analysis (Newton's 2nd law) to solve this.

I've tried using ideas of energy where the Energy at the top=mg0.6+0.5mv2 and Energy at the bottom= 0.5mv2

Give the velocities at top and bottom different symbols and apply conservation of energy.

Also, resolving forces: At the top F=T+mg where F is the centripetal force. At the bottom F=3T-mg.

Good. Apply Newton's 2nd law to top and bottom positions.

That will give you the equations you need to solve for the speeds.