7.7. GRAVITATIONAL POTENTIAL ENERGY

We had discussed earlier the notion of potential energy as being the energy stored in the body at its given position. If the position of the particle changes on account of forces acting on it, then the change in its potential energy is just the amount of work done on the body by the force.

Gravitational potential energy refers to the stored energy of an object based on its height above the ground or its position in a gravitational field. When we lift an object against gravity, we do work on it. The work done on the object is stored as potential energy. Thus, the potential energy changes when the object's position changes due to the force of gravity acting upon it.

Imagine holding a ball in your hand above the ground. The higher you lift the ball, the more work you do on it, and the more potential energy it gains. If you let go, the potential energy converts into kinetic energy as the ball falls.

As we had discussed earlier, forces for which the work done is independent of the path are the conservative forces. The force of gravity is a conservative force and we can calculate the potential energy of a body arising out of this force, called the gravitational potential energy.

Conservative forces allow us to calculate work done in a specific way, where the work depends only on initial and final positions, not the path traveled. Gravity is such a force. The gravitational potential energy can be expressed with the familiar equation W = mgh, where 'W' is the potential energy, 'm' is mass, 'g' is acceleration due to gravity, and 'h' is height above a reference point.

Think of a rollercoaster at the top of a hill. The potential energy of the rollercoaster is at its maximum when it's at the highest point. As it descends, this energy transforms into kinetic energy, leading to a thrilling ride downwards.

Consider points close to the surface of earth, at distances from the surface much smaller than the radius of the earth. In such cases, the force of gravity is practically a constant equal to mg, directed towards the centre of the earth.

Near the Earth's surface, gravity can be considered constant (approximately 9.8 m/s²) for small heights, enabling simpler calculations of potential energy. If you have two heights, h1 and h2, the work done to lift an object from h1 to h2 is calculated directly using the potential difference: W = mg(h2 - h1).

If you’re lifting a backpack from the floor to a table, the force required (weight of the backpack) multiplied by the height difference is the work done against gravity.

If we consider points at arbitrary distance from the surface of the earth, the result just derived is not valid since the assumption that the gravitational force mg is a constant is no longer valid.

Once you move far enough away from Earth, the gravitational force decreases according to the inverse square law. This means the potential energy must be calculated using a different formula involving the mass of the Earth and the distance from its center: W = -G(Mm)/r. This formula shows that potential energy becomes a function of distance from the Earth's center rather than just height.

Think of launching a spacecraft into orbit. As it moves further from Earth, the gravitational pull weakens, and the spacecraft has to reach a specific velocity to maintain its orbit without falling back.

One conventionally sets W1 equal to zero, so that the potential energy at a point is just the amount of work done in displacing the particle from infinity to that point.

In physics, we often define potential energy based on a reference point, often choosing infinity (far away from any gravitational influence) as a zero point for potential energy. This means that the potential energy at a certain point is the energy associated with bringing a mass from infinity to that point.

Imagine climbing a mountain: your potential energy increases as you gain height, and if you were to drop a stone from the top back to the ground, you're sending it back to a point where it possesses no potential energy comparatively.

We have calculated the potential energy at a point of a particle due to gravitational forces on it due to the earth and it is proportional to the mass of the particle. The gravitational potential due to the gravitational force of the earth is defined as the potential energy of a particle of unit mass at that point.

The gravitational potential energy between two masses is determined by their masses and inversely by the distance separating them. The more massive the objects, the stronger the potential energy field they create. This helps in understanding the gravitational interactions not just of Earth with an object, but also between two objects in space.

Consider two people on a seesaw. If one is significantly heavier than the other, the heavier person will have more influence on the seesaw's balance, just like how a larger mass affects gravitational potential energy.

Example 7.3: Find the potential energy of a system of four particles placed at the vertices of a square of side l. Also obtain the potential at the centre of the square.

This example highlights how gravitational potential energy depends on the configuration of multiple particles. By calculating the potential energy based on distances between each pair of particles, students can grasp how energy interacts in a system rather than just between two single masses.

Think of friends lifting each other up in a line: their combined effort (potential energy) can change dramatically based on how close they are to one another and where they are in the line.

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