A tennis ball connected to a string is spun around in a vertical, circular path at a uniform speed. The ball has a mass m = 0.162 kg and moves at v = 4.73 m/s. The circular path has a radius of R = 0.87 m

1)What is the magnitude of the tension in the string when the ball is at the bottom of the circle?
2)What is the magnitude of the tension in the string when the ball is at the side of the circle?
3)What is the magnitude of the tension in the string when the ball is at the top of the circle?
4) What is the minimum velocity so the string will not go slack as the ball moves around the circle?

To find the magnitude of the tension in the string in different positions, we need to consider the forces acting on the tennis ball in each position.

1) When the ball is at the bottom of the circle, the tension in the string needs to provide the centripetal force to keep the ball moving in a circular path. At the bottom, the tension force and the weight of the ball act in the same direction. The equation to calculate the centripetal force is:

Fc = m * v^2 / R

Where:
Fc is the centripetal force
m is the mass of the ball
v is the velocity of the ball
R is the radius of the circular path

Plugging in the values:
Fc = 0.162 kg * (4.73 m/s)^2 / 0.87 m

Calculate Fc to find the magnitude of tension at the bottom of the circle.

2) When the ball is at the side of the circle, the tension force needs to counteract the weight of the ball and provide the centripetal force. The tension force acts horizontally, perpendicular to the weight. In this position, the tension force is equal to the centripetal force. Since the weight acts vertically downward, the tension must be equal in magnitude to the weight.

Calculate the magnitude of the weight:

Weight = m * g

Where:
m is the mass of the ball
g is the acceleration due to gravity (approximately 9.8 m/s^2)

Plugging in the values:
Weight = 0.162 kg * 9.8 m/s^2

The magnitude of tension at the side of the circle is equal to the magnitude of the weight.

3) When the ball is at the top of the circle, the tension force needs to counteract the weight of the ball and provide the centripetal force. In this position, the tension force and the weight of the ball act in opposite directions. The magnitude of the tension can be calculated using the equation:

Fc = m * v^2 / R

However, since the tension force is acting in the opposite direction to the weight, the equation becomes:

Tension = Weight + Fc

Calculate the magnitude of the weight as done in step 2 and plug it into the equation along with the value of Fc calculated using the given values.

4) To find the minimum velocity so the string will not go slack as the ball moves around the circle, we need to consider the tension force at the top of the circle. When the velocity is at its minimum, the tension force must be greater than or equal to zero to prevent the string from going slack. The minimum velocity occurs when the tension force is just enough to counteract the weight of the ball:

Weight = Tension

Rearrange the equation to find the minimum velocity needed:

v = sqrt(R * g)

Calculate the value using the given radius and acceleration due to gravity.