A 4.4-cm-diameter, 24 g plastic ball is attached to a 1.2-m-long string and swung in a vertical circle. The ball's speed is 6.1 m/s at the point where it is moving straight up. What is the magnitude of the net force on the ball? Air resistance is not negligible.

Verified step by step guidance

1. Determine the forces acting on the ball at the top of the vertical circle. These include the gravitational force \( F_g = m g \), the tension in the string \( T \), and the air resistance force \( F_{air} \).
2. Calculate the gravitational force \( F_g \) using \( F_g = m g \), where \( m \) is the mass of the ball (convert 24 g to kg) and \( g \) is the acceleration due to gravity (\( 9.8 \ \text{m/s}^2 \)).
3. Use the centripetal force equation \( F_{net} = \frac{m v^2}{r} \), where \( m \) is the mass of the ball, \( v \) is its speed (6.1 m/s), and \( r \) is the radius of the circle (equal to the length of the string, 1.2 m). This net force is the sum of all forces acting toward the center of the circle.
4. Set up the equation for the net force at the top of the circle: \( F_{net} = T + F_g + F_{air} \). Rearrange to solve for the net force, ensuring all forces are directed toward the center of the circle.
5. Substitute the known values for \( m \), \( g \), \( v \), \( r \), and any given information about air resistance to calculate the magnitude of the net force. Ensure all units are consistent throughout the calculation.

Key Concepts

Centripetal Force

Centripetal force is the net force required to keep 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 centripetal force is provided by the tension in the string and the gravitational force acting on the ball. The formula for centripetal force is F_c = (mv^2)/r, where m is the mass, v is the velocity, and r is the radius of the circular path.

Net Force

The net force on an object is the vector sum of all the forces acting on it. In this case, the net force on the ball includes the gravitational force acting downward and the tension in the string acting upward. To find the net force, one must consider both the centripetal force required for circular motion and the effects of gravity, especially since the ball is moving vertically.

Effects of Air Resistance

Air resistance, or drag, is a force that opposes the motion of an object through air. It is significant in this problem as it affects the net force acting on the ball. The presence of air resistance means that the actual tension in the string must not only provide the centripetal force but also counteract the drag force, altering the calculations for the net force on the ball as it swings in a vertical circle.