Answer:
The tension force in the string at the bottom of the circle is approximately 1.85 N.
Step-by-step explanation:
To solve this problem, we can use the principle of centripetal force, which states that the force required to keep an object moving in a circle is equal to the product of its mass, its velocity squared, and the radius of the circle, divided by the distance from the center of the circle to the object. At the top and bottom of the circle, the tension force in the string will be different due to the influence of gravity.
First, we can find the gravitational force acting on the ball. The weight of the ball can be calculated as:
Fg = mg
where m is the mass of the ball, and g is the acceleration due to gravity.
Substituting the given values, we get:
Fg = (0.04 kg) * (9.81 m/s^2) = 0.3924 N
At the top of the circle, the tension force in the string will be less than the weight of the ball, because gravity is pulling the ball away from the center of the circle, reducing the force required to keep it moving in a circle. Therefore, the tension force can be calculated as:
Ttop = Fg - (m * v^2 / r)
where Ttop is the tension force at the top of the circle, m is the mass of the ball, v is the tangential velocity of the ball, and r is the radius of the circle.
Substituting the given values, we get:
Ttop = 0.3924 N - (0.04 kg * (2 m/s)^2 / 0.22 m) = 0.3924 N - 1.4545 N = -1.0621 N
The negative sign indicates that the tension force is directed upwards, opposite to the direction of the ball's motion. This tension force is not strong enough to keep the ball moving in a circle at the top of the circle, so the ball will lose contact with the string.
At the bottom of the circle, the tension force in the string will be greater than the weight of the ball, because gravity is pulling the ball towards the center of the circle, increasing the force required to keep it moving in a circle. Therefore, the tension force can be calculated as:
Tbottom = Fg + (m * v^2 / r)
where Tbottom is the tension force at the bottom of the circle.
Substituting the given values, we get:
Tbottom = 0.3924 N + (0.04 kg * (2 m/s)^2 / 0.22 m) = 0.3924 N + 1.4545 N = 1.8469 N
Therefore, the tension force in the string at the bottom of the circle is approximately 1.85 N.