A ball having a mass of 4 kg is attached to a string 1 m long and is whirled in a vertical circle at a constant speed of 23 m/s.
(a) Determine the tension in the string when the ball is at the top of the circle.
(b) Determine the tension in the string when the ball is at the bottom of the circle.
T(top ) = m(v²/R – g )
T(bottom ) = m(v²/R + g)
To solve this problem, we can use the concepts of centripetal force and gravitational force.
(a) To determine the tension in the string when the ball is at the top of the circle, we need to consider the forces acting on the ball at that point. At the top, the tension in the string will be the maximum because it needs to provide the necessary centripetal force to keep the ball moving in a circle.
We can start by calculating the centripetal force using the formula:
Fc = (m * v^2) / r
where Fc is the centripetal force, m is the mass of the ball, v is the velocity, and r is the radius of the circle, which in this case is equal to the length of the string.
Plugging in the values, we have:
Fc = (4 kg * (23 m/s)^2) / 1 m
Fc = (4 kg * 529 m^2/s^2) / 1 m
Fc = 2116 kg·m/s^2
Now, since the tension in the string provides the centripetal force, the tension can be considered as equal to the centripetal force at the top of the circle. Therefore, the tension in the string is 2116 kg·m/s^2.
(b) To determine the tension in the string when the ball is at the bottom of the circle, we need to take into account both the centripetal force and the gravitational force acting on the ball. At the bottom of the circle, the tension will be less than the weight of the ball since the tension counteracts only the centripetal force and not the full weight.
To calculate the tension at the bottom of the circle, we can subtract the gravitational force from the centripetal force.
The gravitational force can be calculated using the formula:
Fg = m * g
where Fg is the gravitational force and g is the acceleration due to gravity.
Plugging in the values, we have:
Fg = 4 kg * 9.8 m/s^2
Fg = 39.2 kg·m/s^2
Now, to find the tension at the bottom of the circle, we subtract the gravitational force from the centripetal force:
Tension = Fc - Fg
Tension = 2116 kg·m/s^2 - 39.2 kg·m/s^2
Tension = 2076.8 kg·m/s^2
Therefore, the tension in the string at the bottom of the circle is 2076.8 kg·m/s^2.