A child is swinging a 310-g ball at the end of a 72.0-cm-long string in a vertical circle. The string can withstand a tension of 11.0 N before breaking. What is the tension in the string when the ball is at the top of the circle if its speed at that point is 4.00 m/s?
To find the tension in the string when the ball is at the top of the circle, we need to consider the gravitational force acting on the ball and the centripetal force required to keep it in circular motion.
First, let's find the gravitational force acting on the ball. The weight of an object is given by the equation:
Weight = mass * gravitational acceleration
Given that the mass of the ball is 310 g (which is equivalent to 0.310 kg), and the gravitational acceleration is approximately 9.8 m/s², we can calculate the weight of the ball:
Weight = 0.310 kg * 9.8 m/s² = 3.038 N
The tension in the string is equal to the sum of the centripetal force and the gravitational force. At the top of the circle, the centripetal force is providing the required centripetal acceleration to keep the ball in motion in a circular path.
The centripetal force can be calculated using the equation:
Centripetal force = mass * (velocity² / radius)
Given that the mass of the ball is 310 g (0.310 kg), the velocity is 4.00 m/s, and the radius is the length of the string (72.0 cm), which is equivalent to 0.72 m, we can calculate the centripetal force:
Centripetal force = 0.310 kg * (4.00 m/s)² / 0.72 m = 6.806 N
Finally, we can find the tension in the string by summing the centripetal force and the gravitational force:
Tension = Centripetal force + Weight = 6.806 N + 3.038 N = 9.844 N
Therefore, the tension in the string when the ball is at the top of the circle and the speed is 4.00 m/s is approximately 9.844 N.