1. Derive the formula for potential energy.

Derivation of the Formula for Potential Energy

Introduction to Potential Energy: Potential energy is the energy possessed by an object due to its position relative to other objects. It is a fundamental concept in physics and is crucial in understanding various phenomena, including gravitational and elastic potential energy.

Gravitational Potential Energy: Gravitational potential energy (PEg)(PEg) is the energy stored in an object due to its position in a gravitational field. The formula for gravitational potential energy is derived based on the gravitational force between two objects.

Formula for Gravitational Potential Energy: The formula for gravitational potential energy is given by:

PEg=mghPEg=mgh

Where:

• mm is the mass of the object,
• gg is the acceleration due to gravity (approximately 9.8 m/s29.8m/s2 on the surface of the Earth),
• hh is the height or distance from a reference point.

Derivation of Gravitational Potential Energy Formula: The derivation starts with the definition of work done (WW) against gravity to raise an object to a height hh:

W=F⋅dW=F⋅d

Where:

• FF is the force applied,
• dd is the displacement.

In the case of lifting an object against gravity, the force required is the gravitational force (F=mgF=mg), and the displacement is the vertical height hh.

W=mg⋅hW=mg⋅h

Since work done is equal to the change in potential energy, we can equate it to the change in gravitational potential energy (ΔPEgΔPEg):

ΔPEg=WΔPEg=W

ΔPEg=mghΔPEg=mgh

This is the formula for gravitational potential energy, denoted by PEgPEg.

Conclusion: The formula for potential energy, particularly gravitational potential energy, is derived from the work done against gravity to raise an object to a certain height. It is given by PEg=mghPEg=mgh, where mm is the mass of the object, gg is the acceleration due to gravity, and hh is the height or distance from a reference point. Understanding this formula is essential in various fields of physics and engineering.