A stone is tied to a string (length = 0.450 m) and whirled in a circle at the same constant speed in two different ways. First, the circle is horizontal and the string is nearly parallel to the ground. Next, the circle is vertical. In the vertical case the maximum tension in the string is 18.0% larger than the tension that exists when the circle is horizontal. Determine the speed of the stone.

I know I already asked this question, but how do I get the radius?

To determine the speed of the stone, we need to find the radius of the circular path it follows.

Let's assume that the tension in the string when the circle is horizontal is T, and the maximum tension in the string when the circle is vertical is T_max.

In both cases, the stone is moving in uniform circular motion, which means the gravitational force acting on the stone is balanced by the tension in the string. This can be represented by the following equation:

T = mg (1)

where T is the tension in the string, m is the mass of the stone, and g is the acceleration due to gravity.

When the circle is horizontal, the tension in the string only needs to provide the centripetal force to keep the stone moving in the circle. The centripetal force can be calculated using the following equation:

T = (m * v^2) / r (2)

where v is the speed of the stone and r is the radius of the circular path.

When the circle is vertical, the tension in the string must provide both the centripetal force and counteract the force of gravity. The centripetal force is still given by equation (2), but the gravitational force is now acting downward and needs to be balanced by the tension:

T_max + mg = (m * v^2) / r (3)

We are given that the maximum tension in the vertical case is 18.0% larger than the tension in the horizontal case, so we can write:

T_max = T + 0.18T (4)

Substituting equations (1) and (3) into equation (4), we get:

T + 0.18T = mg + (m * v^2) / r (5)

Since equation (1) tells us that T = mg, we can substitute mg for T in equation (5):

mg + 0.18mg = mg + (m * v^2) / r (6)

Simplifying the equation by canceling out mg on both sides:

0.18mg = (m * v^2) / r (7)

Now, we can rearrange equation (7) to solve for r:

r = (m * v^2) / (0.18mg) (8)

Since we are given the length of the string (0.450 m), we can relate the radius to the length as follows:

r = L - h

where L is the length of the string and h is the height difference between the initial position of the stone and its position at any point in the vertical circle.

To find the height difference h, we can use the fact that the maximum tension in the vertical case occurs at the bottom of the circle, where the stone is at its lowest point. At this point, the tension in the string must balance the gravitational force acting on the stone:

T_max = mg + mgh (9)

Substituting equation (1) and the value of T_max from equation (4) into equation (9), we get:

T + 0.18T = mg + mgh (10)

Simplifying the equation by canceling out the mg on both sides:

T + 0.18T = g + gh (11)

Factoring out T on the left-hand side:

T(1 + 0.18) = g(1 + h) (12)

Simplifying further:

1.18T = g(1 + h) (13)

Simplifying equation (13) in terms of h:

1 + h = (1.18T) / g (14)

h = (1.18T / g) - 1 (15)

Now, we know that the length of the string is equal to the sum of the radius and the height difference:

L = r + h (16)

Substituting equation (8) and (15) into equation (16), we can solve for r:

0.450 m = (m * v^2) / (0.18mg) + ((1.18T / g) - 1) (17)

This equation allows us to solve for the value of r, and then we can use that value to find the speed v using equation (2):

v = sqrt((T * r) / m) (18)

By substituting the given values of T and the mass of the stone into equation (18) along with the calculated value of r from equation (17), we can determine the speed of the stone.