Conservation Of Energy Formula

Conservation Of Energy

Conservation of energy is a fundamental principle in physics that states that the total energy in an isolated system remains constant over time. According to this principle, energy can neither be created nor destroyed; it can only be transformed from one form to another or transferred between objects. This concept is based on the idea that energy is a fundamental quantity that underlies all physical phenomena. The principle of conservation of energy allows for the analysis and understanding of various processes and systems, providing insights into the interplay between different forms of energy and their transformations.

Law of Conservation of Energy

The law of conservation of energy states that energy cannot be created or destroyed in an isolated system. The total amount of energy within a closed system remains constant, and energy can only be transferred or transformed from one form to another.

Conservation of Energy Formula

The conservation of energy is described by the following formula:

\[ E_{\text{initial}} = E_{\text{final}} \]

This equation states that the total energy of a system, represented by E, remains constant over time. The subscripts “initial” and “final” indicate the initial and final states of the system, respectively.

The total energy of a system can be composed of various forms, including:

1. Kinetic Energy (KE): The energy associated with the motion of an object, given by the formula KE = (1/2) x m x v2, where m represents mass and v represents velocity.
2. Potential Energy (PE): The energy associated with the position or configuration of an object relative to a reference point, such as gravitational potential energy (PE_gravity) = m x g x h, where m represents mass, g represents gravitational acceleration, and h represents height.
3. Thermal Energy (TE): The energy associated with the temperature of a system.
4. Chemical Energy: The energy stored in chemical bonds.

Conservation of Energy Formula Derivation

1. Kinetic Energy (KE):

Kinetic energy is the energy possessed by an object due to its motion. For an object of mass m moving with velocity v, the kinetic energy KE is given by:

\[ KE = \frac{1}{2} mv^2 \]

2. Potential Energy (PE):

Potential energy is the energy stored in an object due to its position or configuration relative to other objects. The potential energy PE depends on the type of force acting (e.g., gravitational potential energy, elastic potential energy).

3. Work-Energy Theorem:

According to the work-energy theorem, the work done W on an object is equal to the change in its kinetic energy:

\[ W = \Delta KE \]

If only conservative forces (forces derived from a potential energy function, like gravity or spring force) are involved, the work done by these forces can also change the potential energy of the system.

4. Conservation of Mechanical Energy:

In a closed system where only conservative forces act (no non-conservative forces like friction or air resistance), the total mechanical energy (sum of kinetic and potential energies) remains constant over time. This is known as the conservation of mechanical energy.

Mathematically, for a system with kinetic energy KE and potential energy PE:

\[ KE_i + PE_i = KE_f + PE_f \]

Where:

• KE_i and PE_i are the initial kinetic and potential energies, respectively.
• KE_f and PE_f are the final kinetic and potential energies, respectively.

5. Total Energy Conservation:

The conservation of energy principle extends beyond just mechanical energy to include all forms of energy (such as thermal, chemical, nuclear, etc.). The total energy E of a closed system remains constant:

\[ E_{\text{initial}} = E_{\text{final}} \]

This principle underpins many physical phenomena and is crucial for understanding how energy behaves in various systems, from simple mechanical systems to complex thermodynamic processes.

Solved Examples on Conservation of Energy Formula

Example 1: A 2 kg object is released from a height of 10 meters. Calculate its velocity when it reaches the ground, assuming no air resistance.

Solution:

Given:

Mass of the object (m) = 2 kg

Height (h) = 10 meters

Gravitational acceleration (g) = 9.8 m/s2

Using the conservation of energy, we can equate the potential energy at the initial height to the kinetic energy at the final height:

Potential Energy (PE) at the top = Kinetic Energy (KE) at the bottom

m x g x h = (1/2) x m x v2

(2 kg) x (9.8 m/s2) x (10 m) = (1/2) x (2 kg) x v2

v2 = (2 kg x 9.8 m/s2 x 10 m) / 2 kg

v2 = 98 m2/s2

v ≈ 9.90 m/s

Therefore, the velocity of the object when it reaches the ground is approximately 9.90 m/s.

Example 2: A pendulum of mass 0.5 kg is released from a height of 1 meter. Determine its maximum height reached on the other side of its swing.

Solution:

Given:

Mass of the pendulum (m) = 0.5 kg

Height (h) = 1 meter

Gravitational acceleration (g) = 9.8 m/s2

Using the conservation of energy, we can equate the potential energy at the initial height to the potential energy at the maximum height:

Potential Energy (PE) at the top = Potential Energy (PE) at the maximum height

m x g x h = m x g x h_max

(0.5 kg) x (9.8 m/s2) x (1 m) = (0.5 kg) x (9.8 m/s2) x h_max

h_max = 1 meter

Therefore, the maximum height reached by the pendulum on the other side of its swing is 1 meter.

Conclusion

The conservation of energy principle allows for the interconversion of energy between different forms. For example, potential energy can be converted to kinetic energy, and vice versa, without a net change in the total energy of the system, as long as no external forces or non-conservative forces are acting.

This principle is widely applied in various fields of physics, such as mechanics, thermodynamics, and electromagnetism, to analyze and understand the behavior of systems and to make predictions about their energy transformations.