A 100 g ball on a 60-cm-long string is swung in a vertical circle about a point 200 cm above the floor. The string suddenly breaks when it is parallel to the ground and the ball is moving upward. The ball reaches a height 600 cm above the floor. What was the tension in the string an instant before it broke?

Verified step by step guidance

1. Step 1: Identify the forces acting on the ball just before the string breaks. These include the tension in the string and the gravitational force acting downward. The ball is moving in a circular path, so there is also a centripetal force required to maintain circular motion.
2. Step 2: Use the principle of conservation of energy to determine the ball's velocity at the moment the string breaks. The ball's kinetic energy at this point is converted into potential energy as it rises to its maximum height. Write the energy conservation equation: \( KE_{initial} + PE_{initial} = PE_{final} \), where \( KE = \frac{1}{2}mv^2 \) and \( PE = mgh \).
3. Step 3: Solve for the velocity \( v \) of the ball at the moment the string breaks using the energy conservation equation. Substitute \( m = 0.1 \, \text{kg} \), \( h_{initial} = 2.6 \, \text{m} \) (sum of the string length and height above the floor), and \( h_{final} = 6 \, \text{m} \).
4. Step 4: Apply Newton's second law in the radial direction to find the tension in the string. The net radial force is the sum of the tension and the gravitational force component acting along the radial direction. Write the equation: \( T - mg = \frac{mv^2}{r} \), where \( r = 0.6 \, \text{m} \) is the radius of the circular motion.
5. Step 5: Rearrange the equation to solve for the tension \( T \): \( T = mg + \frac{mv^2}{r} \). Substitute the values for \( m \), \( g = 9.8 \, \text{m/s}^2 \), \( v \) (calculated in Step 3), and \( r \) to find the tension in the string just before it broke.

Key Concepts

Centripetal Force

Centripetal force is the net force acting on an object moving in a circular path, directed towards the center of the circle. In this scenario, the ball is undergoing circular motion, and the tension in the string provides the necessary centripetal force to keep the ball moving along its circular path. The tension must counteract both the gravitational force acting on the ball and provide the required centripetal force.

Gravitational Potential Energy

Gravitational potential energy (GPE) is the energy an object possesses due to its position in a gravitational field, calculated as GPE = mgh, where m is mass, g is the acceleration due to gravity, and h is height. In this problem, understanding how the ball's height changes after the string breaks is crucial for determining the energy transformations that occur as it moves upward and reaches a maximum height.

Energy Conservation

The principle of energy conservation states that energy cannot be created or destroyed, only transformed from one form to another. In this context, the kinetic energy of the ball at the moment the string breaks is converted into gravitational potential energy as it rises. Analyzing the energy conservation allows us to relate the ball's speed and height to find the tension in the string just before it broke.