Law of Conservation of Energy: Definition Formula Derivation And Example

What is the law of conservation of energy? The Law of Conservation of Energy is a fundamental concept in physics that states that energy cannot be created or destroyed in an isolated system; it can only be transformed from one form to another. This principle is useful in understanding the behaviour of energy in various physical processes and is essential in fields such as mechanics, thermodynamics, and electromagnetism.

Law of Conservation of Energy Definition

The Law of Conservation of Energy states that the total energy of an isolated system remains constant over time. In other words, the total amount of energy in a closed system is conserved, although it may change forms, such as from kinetic energy to potential energy, thermal energy, or other forms of energy.

Law of Conservation of Energy Meaning

This principle implies that energy can be transferred between objects or converted into different forms, but the total energy within the system remains unchanged. For example, when a pendulum swings, its energy continuously converts between kinetic energy (movement) and potential energy (height). Still, the sum of these energies remains constant if we neglect air resistance and friction.

Law of Conservation of Energy Formula

The general formula for the conservation of energy in a closed system can be expressed as U_T = U_i + W + Q where;

• U_T: Total energy of the system at a specific time.
• U_i: Initial energy of the system before any transformation occurs.
• W: Work done on or by the system (energy transfer due to forces acting on the system).
• Q: Heat transferred into or out of the system (energy transfer due to temperature difference).

OR

The general formula for the conservation of energy in a closed system can be expressed as:

E_total=constant

For mechanical systems, this can be written as:

K+U=constant

Where:

• K = Kinetic energy
• U = Potential energy

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Derivation of Conservation of Energy

Let's derive U_T=U_i+W+Q:

1. Initial Total Energy: U_i

2. Work Done on the System: W

3. Heat Added to the System: Q

4. Change in Internal Energy: ΔU=U_T−U_i

5. First Law of Thermodynamics: ΔU=W+Q

6. Combining: U_T−U_i=W+Q

7. Rearranging: U_T=U_i+W+Q

This equation represents the Law of Conservation of Energy, where U_T​ is the total energy, U_i is the initial energy, W is the work done, and Q is the heat added.

OR

• Kinetic Energy (K)

The kinetic energy K of a particle with mass m and velocity v is given by: K=1/2mv²

• Potential Energy (U)

The potential energy U associated with a conservative force can be expressed as a function of position. For example, in a gravitational field near the Earth's surface, the potential energy is: U=mgh where h is the height above a reference level.

• Work-Energy Theorem

The work-energy theorem states that the work done by the net force on a particle is equal to the change in its kinetic energy: W=ΔK

For a conservative force, the work done W is also related to the change in potential energy: W=−ΔU

This implies: ΔK=−ΔU

• Total Mechanical Energy

The total mechanical energy E of the system is the sum of the kinetic and potential energies: E=K+U

• Conservation of Mechanical Energy

From the work-energy theorem and the relationship between work and potential energy, we have: ΔK=−ΔU

Rearranging this, we get: ΔK+ΔU=0

This implies that the change in the total mechanical energy is zero: Δ(K+U)=0

Thus, the total mechanical energy remains constant: K+U=constant

Law of Conservation of Energy Examples

Example 1: Pendulum

A pendulum of mass mmm swings from a height h. At the highest point, the pendulum has maximum potential energy and zero kinetic energy. At the lowest point, it has maximum kinetic energy and zero potential energy.

• At height h:

U=mgh

K=0

E_total=mgh

• At the lowest point:

U=0

K=1/2mv²

E_total=1/2mv²

Since energy is conserved: mgh=1/2mv²

Example 2: Roller Coaster

A roller coaster starts at the top of a hill with height h and potential energy U=mgh. As it descends, potential energy converts to kinetic energy.

• At the top of the hill:

U=mgh

K=0

E_total=mgh

• At the bottom of the hill: U=0

K=1/2mv²

E_total=1/2mv²

• Since energy is conserved:

mgh=1/2mv²

Law of Conservation of Energy Applications

1. Engineering: Understanding energy conservation is crucial in designing engines, machines, and structures.
2. Renewable Energy: Helps in analysing the conversion of energy forms, such as from solar to electrical energy.
3. Environmental Science: Used to study energy flow in ecosystems and the impact of human activities on energy resources.
4. Astrophysics: Essential in understanding the dynamics of celestial bodies and the universe's energy balance.

References and Further Reading

You can refer to the reference material provided in the below mentioned links for better preparation and understanding scientific concepts.

• Books

NCERT Class 10 Science Textbook PDF

NCERT Class 12 Physics Textbook PDF

• Articles

Magnetic effects of Electric Current

• Online Resources

CBSE Class 10 Science Video Tutorials

CBSE Class 12 Physics Video Tutorials

Also Check:

• CBSE Class 10 Science Syllabus 2024-25: Download PDF
• CBSE Class 12 Syllabus 2024-25 PDF (All Subjects)
• CBSE Class 11 Syllabus 2024-25: Latest and Revised PDF