A uniform metal ball (mass m) is tied to the end of a string (massless, length L). The other end of the

string is fixed. The ball is moving in a vertical circle centered at the fixed end of the string. Suppose
the velocity of the ball at the lowest point is v, what is the string tension when it makes an angle x to
the downward vertical position?

To find the string tension when the ball makes an angle x to the downward vertical position, we can use the principles of centripetal force and gravitational force.

At the lowest point of the ball's motion, the tension in the string is equal to the sum of the gravitational force acting on the ball and the centripetal force required to keep the ball moving in a circle.

1. Centripetal force: The centripetal force is the force that acts towards the center of the circular motion, keeping the ball in its curved path. In this case, it is provided by the tension in the string.

The formula for centripetal force is given by:

F_c = (mv^2) / R

where F_c is the centripetal force, m is the mass of the ball, v is the velocity of the ball at the lowest point, and R is the radius of the circular motion (which is equal to the length of the string, L).

2. Gravitational force: The gravitational force is the force that acts vertically downward on the ball due to gravity. It can be calculated using the mass of the ball and the acceleration due to gravity.

F_g = mg

where F_g is the gravitational force, m is the mass of the ball, and g is the acceleration due to gravity (approximately 9.8 m/s^2).

3. Tension force: The tension in the string is the force that acts along the string, balancing the centripetal force and the gravitational force.

At an angle x to the downward vertical position, we can break down the tension force into two components:

- The horizontal component of tension, T_h, which is equal to F_c.
- The vertical component of tension, T_v, which is equal to the sum of F_g and the vertical component of F_c.

Using trigonometry, we can find the values of T_h and T_v:

T_h = F_c = (mv^2) / R

T_v = F_g + F_c * sin(x)

Finally, the tension in the string, T, can be found by using the Pythagorean theorem:

T = sqrt(T_h^2 + T_v^2)

Substituting the values we obtained earlier, we can find the string tension when the ball makes an angle x to the downward vertical position.