Problem: A stone is rotated in a vertical circle. Speed at bottommost point is √8gR, where R is the radius of the circle. The ratio of tension at the top and the bottom is:

Solution:
To solve this problem, we need to apply the laws of centripetal force, gravity, and tension. We assume that the stone is rotating in a vertical circle of radius R. The speed of the stone at the bottommost point is given as √8gR, where g is the acceleration due to gravity. The ratio of tension at the top and the bottom can be found as follows:

Step 1: Find the velocity of the stone at the topmost point
At the topmost point, the velocity of the stone is zero. We can use the conservation of energy principle to find the velocity of the stone at the topmost point. The total energy of the stone is given as:

E = mgh + 1/2mv^2

where m is the mass of the stone, h is the height of the topmost point, and v is the velocity of the stone at the topmost point.

At the bottommost point, the total energy of the stone is:

E = mgh + 1/2mv^2 = mgh + 1/2m(√8gR)^2 = mgh + 4mgR

Since the total energy of the stone is conserved, we can equate the energy at the topmost point to the energy at the bottommost point:

mgh = mgh + 4mgR + 1/2mv^2

Solving for v, we get:

v = √(2gR)

Step 2: Find the tension at the topmost point
At the topmost point, the tension in the string is equal to the weight of the stone minus the centripetal force required to keep the stone moving in a circle. The weight of the stone is given as:

W = mg

The centripetal force required to keep the stone moving in a circle of radius R is:

F = mv^2/R

Substituting the values of m and v, we get:

F = m(2gR)/R = 2mg

Therefore, the tension at the topmost point is:

T1 = mg - F = mg - 2mg = -mg (upward)

Step 3: Find the tension at the bottommost point
At the bottommost point, the tension in the string is equal to the weight of the stone plus the centripetal force required to keep the stone moving in a circle. The weight of the stone is given as:

W = mg

The centripetal force required to keep the stone moving in a circle of radius R is:

F = mv^2/R

Substituting the values of m and v, we get:

F = m(8gR)/R = 8mg

Therefore, the tension at the bottommost point is:

T2 = mg + F = mg + 8mg = 9mg (downward)

Step 4: Find the ratio of tension at the top and bottom
The ratio of tension at the top and bottom is:

T1/T2 = (-mg)/(