Problem 15 - University Physics with Modern Physics

Question

A girl of mass \(49.0 \mathrm{~kg}\) is on a swing, which has a mass of \(1.0 \mathrm{~kg} .\) Suppose you pull her back until her center of mass is \(2.0 \mathrm{~m}\) above the ground. Then you let her \(\mathrm{go},\) and she swings out and returns to the same point. Are all forces acting on the girl and swing conservative?

Step-by-Step Solution

Answer

Answer: Yes, all the forces acting on the girl and the swing are conservative. The gravitational force is conservative, and the tension force does no work on the girl and the swing.

Step 1: Consider the gravitational force

The gravitational force is a conservative force. This is because the work done by the gravitational force on an object only depends on the object's initial and final heights, not the path taken. In this problem, the girl and swing move from an initial height of 2.0 meters to a final height of 2.0 meters, so the work done by the gravitational force is the same regardless of their path.

Step 2: Consider the tension force

The tension force is the force exerted by the swing's rope on the girl and the swing as they move. This force always acts perpendicular to the path of the girl and the swing. Since the work done by a force is the product of the force and the distance moved along the path, and the angle between the force and the path, the work done by the tension force is zero (0 J) for any path taken by the girl and the swing since the angle between the tension force and the path is always 90 degrees.

Step 3: Determine if all forces are conservative

Since the gravitational force is conservative and the tension force does no work on the girl and the swing, we can conclude that all forces acting on the girl and the swing are conservative.

Key Concepts

Gravitational Force

Gravitational force is a fundamental concept in physics and is one of the key conservative forces. It acts on any object with mass and pulls it towards the center of the Earth. What's important to understand is that gravitational force only depends on the vertical distance through which an object is moved.

- A force that only depends on initial and final positions, not the path taken. - Works no matter the direction of motion. - Always acts vertically downwards toward Earth's core.
For the swing, the gravitational force was responsible for pulling the girl down after being pulled to the height of 2 meters. Since she ended up at the same height on her return swing, gravitational work sums to zero. This indicates that energy conservation holds, aligning with the principle that gravitational force is a conservative force.

Tension Force

Tension force is usually observed in strings, ropes, or cables that are under strain. In the scenario with the swing, the rope exerts this force, helping to sustain the circular motion of the swing.

Tension Force Characteristics:

• Always acts along the line of the rope and perpendicular to the swing's path.
• Does not perform work because it acts at a 90° angle to the direction of motion.

Since work is calculated as the product of force and distance in the direction of force, and given that tension is perpendicular to this direction, no work is done. The lack of work indicates that the tension force does not contribute to energy change in the girl and swing system. It merely changes the direction of motion, adding support to its conservative nature.

Work-Energy Principle

The work-energy principle is a fundamental concept that connects the dots between force, work, and energy. It states that the work done by all forces on an object equals the change in its kinetic energy. In essence, it converts the concepts of force and motion into an energy transformation perspective.

Key Points about Work-Energy Principle:

• Total work done by conservative forces equals the change in potential energy.
• No net energy loss where only conservative forces apply, like gravity, due to perfect energy transformation back and forth.

In our swing example, since both gravitational and tension forces are respective conservative forces, the swing's potential energy at its highest point is converted entirely back to kinetic energy and again to potential. This energy conversion without loss validates the work-energy principle, demonstrating a consistent and cyclical energy transfer in systems dominated by conservative forces.