Why Is Tension at the Lowest Point of a Vertical Circle Significantly Higher?

[Karol]

Homework Statement

A stone tied to a rope rotates in a vertical circle. prove that the tension in the rope at the lowest point is 6 times the stone's weight bigger than at the highest point.

Homework Equations

Potential energy: $E_P=mgh$
Kinetic energy: $E_K=\frac{1}{2}mV^2$
Radial force: $F_R=m\frac{V^2}{R}$

The Attempt at a Solution

V₀ is the velocity at the top and V₂ is at the bottom and R is the radius.
$$\frac{1}{2}mV_0^2=\frac{1}{2}mV_2^2-2Rmg \rightarrow V_2^2=V_0^2+4gR$$
The ratio of radial forces at the bottom and at the top:
$$\frac{F_B}{F_T}=\frac{\frac{V_B^2}{R}}{\frac{V_T^2}{R}}=\frac{V_B^2}{V_T^2}=\frac{V_0^2+4gR}{V_0^2}=1+\frac{4gR}{V_0^2}$$
First it includes V₀ and R, it's not fixed, and secondly it doesn't even come close to the form.
Of course i have to deduce, at the upper point, the weight of the stone from the radial force and add it at the lowest point, but my solution doesn't even come close.

 

[haruspex]

You are asked to consider the difference in the two forces, not the ratio.
Also, don't forget the force of gravity on the stone. How will that affect the two tensions?

 

[dauto]

You calculated the ratio between the radial forces. You want the ratio between the tensions. Not the same thing.

 

[haruspex]

You want the ratio between the tensions.

No, Karol does not want the ratio of the tensions. The question refers to the ratio between the stone's weight and the difference between the tensions.