What is Potential Energy?
Potential energy is the energy possessed by an object due to its relative stationary position in space, stress, or electric charge. Potential energy is the inherent energy of the body relative to its static position to the other objects. Potential energy is one of the two main types of energy, while the other is kinetic energy. The two types of potential energy are elastic potential energy and gravitational potential energy.
Elastic Potential Energy: Elastic Potential Energy is the energy present in objects that can be stretched or extended, such as trampoline, rubber bands, and bungee cords. The further an object can expand, the more elastic potential energy it has. Many items are designed specially to store elastic potential energy such as a twisted rubber band that powers a toy plane or a Coil spring of a wind-up clock.
The elastic potential energy formula derivation is:
U = 1/2 kx2
Where,
U = elastic potential energy
k = spring force constant
x = string stretch length in m
Gravitational Potential Energy: Gravitational potential energy is the energy acquired by an object due to a shift in its position when it is present in a gravitational field. In simple terms, it can be stated that gravitational potential energy is an energy that is linked to gravity or gravitational force.
The gravitational potential energy equation is:
GPE = m × g × h,
m = mass in kilograms,
g = acceleration (9.8 ms-2 on Earth)
h = height.
Derivation for Potential Energy
The derivation of potential energy is discussed here. Potential energy is determined as the energy that is held by an object because of its stationary position. Joule is the S.I. unit of potential energy; its symbol is J. Scottish engineer and physicist William Rankine coined the term potential in the 19th century. The potential energy formula depends on the force enacting on two objects. The formula of gravitational potential energy is:
W = m × g × h = mgh
m = mass in kilograms
g = acceleration due to gravity
h = height in meters.
Gravitational Potential Energy Derivation Equation
Let us consider an object, of mass M, which is placed along the x-axis, and there is a test mass m at infinity. Work done at bringing it without acceleration through a minimal distance (dx) is given by:
dw = Fdx
Here, F is an attractive force and towards the negative x-axis direction is the displacement. Therefore, F and dx are in a similar direction.
\[dw = (\frac{GMm}{x^{2}}) dx\]
Integrating both sides,
\[w = \int_{r}^{\infty} \frac{GMm}{x^{2}} dx\]
\[w = -[ \frac{GMm}{x}]\]
\[w = -[\frac{GMm}{r}] - (\frac{-GMm}{\infty})\]
\[w = \frac{-GMm}{r}\]
As the potential energy is stored as U, the gravitational potential energy at ‘r’ distance from the object having mass ‘M’ is:
U = - GMm/r
Now if another mass inside the gravitational field moves from one point inside the field to another point of the field of mass M, the other mass experiences a change in potential energy given by:
ΔU = GMm (\[\frac{1}{r_i} – \frac{1}{r_f}\]) ( ri= initial position and rf= final position )
If ri > rf then ΔU is negative.
Derive an Expression for Gravitational Potential Energy at Height ‘h’
Let‘s consider an object taken to a height ‘h’ from the surface of the earth.
ri = R and rf = R + h
then,
ΔU = GMm \[\frac{1}{R} – \frac{1}{(R+h)}\]
ΔU = GMmh/R(R + h)
When, h<<R, then, R + h = R and g = GM/R2.
On substituting this in the above equation we get,
Gravitational Potential Energy ΔU = mgh.
FAQs on Derivation of Potential Energy
1. What is potential energy and what are its main types as per the CBSE syllabus?
Potential energy is the stored energy an object possesses due to its position, state, or configuration. It is a form of mechanical energy. The two primary types of potential energy studied are:
- Gravitational Potential Energy: The energy stored in an object due to its position in a gravitational field, such as a book held above the ground.
- Elastic Potential Energy: The energy stored in an elastic object when it is stretched or compressed, like a drawn bow or a compressed spring.
2. How do you derive the simple formula for gravitational potential energy, U = mgh?
The derivation for gravitational potential energy near the Earth's surface is based on the work done against gravity. Consider an object of mass m lifted to a height h.
- The force required to lift the object against gravity is equal to its weight, F = mg, where 'g' is the acceleration due to gravity.
- Work done (W) is calculated as the product of force and displacement (height), so W = F × h.
- Substituting the force, we get W = (mg) × h = mgh.
- This work done is stored in the object as its gravitational potential energy (U). Therefore, U = mgh.
3. What is the derivation for elastic potential energy stored in a spring?
The derivation for elastic potential energy relies on Hooke's Law. When a spring is stretched or compressed by a distance x from its equilibrium position, it exerts a restoring force, F = -kx, where 'k' is the spring constant. The work done to stretch the spring is stored as potential energy.
- The work done (dW) for a small displacement (dx) is dW = F_applied * dx = kx dx.
- To find the total work done in stretching from 0 to x, we integrate this expression: W = ∫(kx) dx from 0 to x.
- Solving the integral gives W = k [x²/2] from 0 to x, which simplifies to W = ½ kx².
- This work done is the elastic potential energy (U) stored in the spring, so U = ½ kx².
4. How is the general formula for gravitational potential energy, U = -GMm/r, derived using calculus?
The general formula for gravitational potential energy is derived by calculating the work done to bring a mass m from infinity to a point r in the gravitational field of a larger mass M. The zero potential energy level is conventionally set at infinity.
- The gravitational force between the masses is F = GMm/x², where x is the variable distance.
- The work done (dW) to move the mass 'm' by a small distance 'dx' against this force is dW = F dx.
- The total work done, which equals the potential energy, is the integral of the force from infinity (∞) to the point r: U = ∫(GMm/x²) dx from ∞ to r.
- Solving the integral gives U = -[GMm/x] from ∞ to r.
- This results in U = -GMm/r - (-GMm/∞), which simplifies to U = -GMm/r. The negative sign indicates that the force is attractive.
5. What is the physical significance of the relationship F = -dU/dx?
The equation F = -dU/dx represents a fundamental relationship between a conservative force (F) and its associated potential energy (U). It states that the conservative force is the negative gradient of the potential energy. The physical significance is that the force on an object always points in the direction where its potential energy decreases most steeply. For example, a ball on a hill (high potential energy) experiences a gravitational force that pushes it towards the bottom of the hill (low potential energy).
6. How does the derivation of electric potential energy differ from gravitational potential energy?
While both derivations involve calculating the work done to move an object in a force field, the key difference lies in the nature of the force.
- Gravitational Potential Energy is derived using Newton's Law of Universal Gravitation (F = GMm/r²), which is always an attractive force.
- Electric Potential Energy is derived using Coulomb's Law (F = kq₁q₂/r²). This force can be either attractive (between opposite charges) or repulsive (between like charges).
- Consequently, electric potential energy can be positive (for repulsive forces) or negative (for attractive forces), whereas gravitational potential energy in the U=-GMm/r form is always negative because gravity is always attractive.
7. Why is potential energy always defined for a system of objects rather than a single object?
Potential energy is fundamentally an interaction energy. It arises from the forces between two or more objects. A single, isolated object cannot have potential energy. For example:
- Gravitational potential energy exists because of the interaction between the Earth and an object. Without the Earth, the object would not have this energy.
- Elastic potential energy exists because of the interaction between the different parts of a spring or an elastic body as they are deformed. It's the energy of the system (e.g., spring and the deforming hands), not just the spring in isolation.
- Therefore, potential energy is a property of the system as a whole.
