A ball of mass 50 g tied to the end of a 50 cm light and inextensible string is whirled around in a vertical circle. Find the tension in the string when the ball is at the top of the circle. [Take g=10 ms-2]
To find the tension in the string when the ball is at the top of the circle, we need to analyze the forces acting on the ball.
At the top of the circle, the tension in the string provides the centripetal force that keeps the ball moving in a circle. In addition, we have the gravitational force acting downward.
1. Centripetal Force: F_c = m * v^2 / r
In this case, the velocity, v, can be determined by considering the tension in the string to be the centripetal force:
T = m * v^2 / r
2. Gravitational Force: F_g = m * g
At the top of the circle, the gravitational force acts in the opposite direction of the tension in the string.
Since the forces are in opposite directions, the net force is the difference between the two forces:
net force = F_c - F_g = m * v^2 / r - m * g
Since the ball is at the top of the circle, the net force must be directed towards the center of the circle, which means it must be positive. Therefore, the net force is equal to the centripetal force:
m * v^2 / r - m * g = m * v^2 / r
Simplifying the equation:
m * v^2 / r = m * g
v^2 = g * r
v = √(g * r)
v = √(10 * 0.50) = √5 m/s
Now, we can substitute the value of v into the equation for the tension:
T = m * v^2 / r
T = (0.05 kg) * (5 m/s)^2 / (0.50 m)
T = 0.05 m * 25 m^2/s^2 / 0.50 m
T = 1 kg m/s^2 = 1 N
The tension in the string when the ball is at the top of the circle is 1 Newton (N).