The **work** done by a force on a particle is defined as the product of the component of the force in the direction of the displacement and the magnitude of the displacement. Suppose a constant force **F** acts on a block to produce a displacement **s** in it. The angle contained between the vectors **F** and **s** is θ . Then the work done by the force is:

`W = Fs cos θ`

Work is the scalar product of force and displacement vectors, which means work itself is a scalar and has no direction. If the angle between **F** and **s** is an acute angle (< 90°), work is positive. If the angle between **F** and **s** is an obtuse angle (> 90°), work is negative. If the angle between F and **s** is 90° , work done is zero. When a block moves on a horizontal floor, the force of gravity exerted by the earth and the normal force exerted by the floor do not do any work on the block because these forces act at right angles to the motion of the block. The dimensions of work are `[ML`

, and its SI unit is joule (J).
^{2}T^{-2}]

To calculate the **work done by a varying force**, we need a more rigorous, calculus-based approach. Let us consider the case of a **block attached to a spring**. Suppose one end of a spring is attached to a fixed wall and the other end is attached to a block resting on a horizontal, frictionless floor. Initially, the spring is at its natural length. Now if an external force is applied to the block to move it slowly without acceleration further away from the wall, the spring stretches and exerts a variable restoring force opposite to the block's motion. The magnitude of this **spring force** is directly proportional to the displacement of the block.

The constant of proportionality is known as the **force constant** of the spring. If the spring is stretched to take the block from initial position x = 0 to final position x = x, it can be shown by integration that the **work done by the external force** is:

`W`

_{ext}= ^{1}/_{2} kx^{2}

The corresponding **work done by the spring force** is:

`W`

_{spr}= -^{1}/_{2} kx^{2}

Here k is the force constant of the spring. Notice that the external force does positive work because it has the same direction as the block's displacement. The spring force does negative work (of the same absolute value) because it is directed opposite to the block's displacement.

The **energy** of a body can be loosely defined as its capacity of doing work, measured by the total amount of work the body can do. Energy can be of different types, e.g. mechanical, thermal, light, electrical, magnetic, sound, chemical and nuclear. In classical mechanics, we shall mostly deal with **mechanical energy** which is further classified into two types: **kinetic energy** associated with the motion of a body, and **potential energy** associated with the configuration of a system such as gravitational, elastic and electrical potential energy. All types of energy have the same dimensions and units as those of work.

The **kinetic energy** K of a particle of mass m moving with speed v is defined as:

`K= `

^{1}/_{2} MV^{2}

This is a scalar quantity which is always positive. According to the **work-energy theorem**, the net work done on a particle is equal to the change in its kinetic energy. In mathematical terms:

```
W
```

_{net} = ^{1}/_{2} mv_{2}^{2} - ^{1}/_{2} mv_{1}^{2} = K_{2} - K_{1} = ΔK

where v_{1} and v_{2} are the initial and final speeds, K_{1} and K_{2} are the initial and final kinetic energies, Δ K is the change in kinetic energy of the particle.

The **power** supplied by a force is defined as the rate at which the force does work. It is the scalar product of force and velocity vectors. That is:

```
P =
```

**F**.**v** = Fv cos θ

where θ is the angle contained between **F** and **v**. The dimensions of power are `[ML`

, and its SI unit is watt (W).
^{2}T^{-3}]

The **potential energy** of a system is the energy possessed by it by virtue of its configuration. It is measured by the work the system can do if it returns from its present configuration to some standard configuration usually associated with zero potential energy.

Forces in nature can be divided into two groups: **conservative force** and **non-conservative force**. A force is conservative if the total work done by it on a particle moving in a closed path is zero. In other words, the work done by a conservative force on a particle is independent of the path it takes to move from one point to another. Examples: force of gravity, spring force.

The **energy diagram** is the graph that shows the variation in the potential energy U(x) of a system with the displacement x of a particle forming part of the system. Energy diagram helps us to determine whether a body is in **stable equilibrium**, **unstable equilibrium** or **neutral equilibrium**.

According to the **principle of conservation of mechanical energy**, the total mechanical energy of an isolated system of particles which interact only through conservative forces is constant. If K_{1} , U_{1} are the initial values and K_{2} , U_{2} are the final values of the kinetic and potential energies of the system then:

`K`

_{1} + U_{1} = K_{2} + U_{2}

The principle can be applied to solve a host of mechanical problems. The method is often referred to as the **energy method** of solving a problem. It is usually more elegant and easier than force-based **dynamical method** of solving a problem.

Another conservation principle with broader scope is the **principle of conservation of energy**. It states that energy can neither be created nor destroyed. It may be transformed from one form to another, but the total energy of an isolated system is always constant. From a universal point of view, the total energy of the universe is constant.

Any discussion on energy conservation remains incomplete without reference to Einstein's famous formula on **equivalence of mass and energy: **

```
E
```

_{0} = m_{0} c^{2}

In this expression, m_{0} is the **rest mass** of a particle, E_{0} is the **rest energy** which is energy equivalent of m_{0} , and c is the speed of light in vacuum.

Concept Of Energy And Derivation Of Work-Energy Theorem
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Conservative And Non-Conservative Forces
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Conservative Force From Potential Energy Function, And Energy Diagrams
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Definition Of Work
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Definition Of Work II
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Equivalence Of Mass And Energy
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More Applications Of Mechanical Energy Conservation Principle
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Potential Energy
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Potential Energy Function
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Power
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Principle Of Conservation Of Energy, And Some Applications
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Principle Of Conservation Of Mechanical Energy, And Some Applications
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Problems On Conservative And Non-Conservative Forces
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Problems On Kinetic Energy And Work-Energy Theorem
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Problems On Principle Of Mechanical Energy Conservation
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Work Done By A Varying Force
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Work Done By Spring Force
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