Gravitational Potential Energy of a System

Gravitational potential energy is the energy that an object has due to its position in a gravitational field. An item at the Earth’s surface with a constant gravitational acceleration of 9.8 m/s² is the most common application of gravitational potential energy. Since gravitational potential energy – zero can be chosen anywhere (just the same as coordinate system zero), the potential energy at a height h beyond that point is equal to the work necessary to lift the item to that height while maintaining no net change in kinetic energy.

The gravitational potential energy is equal to the weight times the height it is lifted since the force necessary to hoist it is similar to its weight.

                            PEg = weight * height = mgh

Gravitational potential 

The gravitational potential at that place is equal to the work (energy communicated) per unit mass necessary to move an object to that position from a fixed reference site. It’s similar to the electric potential, but instead of charge, mass is used. According to the convention, the potential’s reference position is infinitely far from any mass, resulting in a negative potential at any finite distance.

Change in gravitational potential energy

The work done by the gravitational force is related to the change in a system’s gravitational potential energy. We will create an expression to determine the change in potential energy for a system of two particles in this section. We examine a simple set-up consisting of a fixed mass particle, “m1,” and a moving mass particle, “m2,” which moves from one place to another for this. Now we know that the change in the system’s potential energy is equal to the work done by gravitational force for the displacement of the second article:

                                ΔU=−WG

Work done by gravitational force, on the other hand, is given as:   

                            WG= r2∫r1 FGdr

The mathematical statement for determining the change in potential energy of the system is produced by combining two equations as follows:

                            ΔU=− r2∫r1FGdr

We must first set up the differential equation to evaluate this integral. We suppose that the stationary particle is located at the reference origin for this. Furthermore, we consider a mass “m2” particle in a position halfway between two straight-line positions. As the particle goes from location “r” to “r+dr,” the system’s potential energy changes to

                              F=−Gm1m2/r²

Integrating between the initial and final positions of the second particle yields the following formula for the change in gravitational potential energy:

                           ⇒ΔU= r2∫r1Gm1m2dr/r²

We have by removing constants from the integral and integrating between the limits.      

                           ΔU = U2 – U1 = Gm1m2(r1 -r2)/r1r2

When a particle of mass “m2” moves from “r1” to “r2” in the presence of a particle’s mass “m1 “, this is the expression of gravitational potential energy change. It’s crucial to remember that “r1” is more significant than “r2 “. This example signifies that the change in gravitational potential energy is positive. To put it another way, it means that the final value is higher than the initial value. As a result, the gravitational potential energy of the two-particle system increases as the linear distance between them increases.

Examples of the gravitational potential energy of a system

The gravitational potential energy of anything in a high position is high. A book on the top shelf, for example, has more potential energy than a book on the bottom shelf because it has a longer distance to fall. Other things having gravitational potential energy include the following:

• Object lifted to a certain height
• A car parked on a hilltop.
• Yo-yo is on the verge of being released
• A book on the verge of slipping off a table
• A bird is resting on a tree branch.
• A giant ball of a demolishing machine
• The water behind the dam wall
• Aeroplane flying overhead

Conclusion

Gravitational potential energy is the energy that an object has because of its location in the gravitational field. An item at the Earth’s surface with a constant gravitational acceleration of 9.8 m/s2 is the most common application of gravitational potential energy. The gravitational potential at that place is equal to the work (energy communicated) per unit mass necessary to move an object to that position from a fixed reference site. The gravitational force’s work is proportional to a system’s gravitational potential energy change.