E1.5 Work, Energy and Power (Potential Energy):

Energy is a central concept in Physics. It can be defined as the capacity for doing work. But, it is important to understand that just because energy exists, not necessarily available to do work.

Energy can not be create or destroyed, but it can change from one form to another form.

There are many types of energy such as: Mechanical energy, chemical energy, light energy, thermal energy, electrical energy etc. Mechanical energy further divided into two parts called: Kinetic Energy and Potential Energy. Here, we mainly focused on these two energy.

 

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Potential Energy:

"Energy that is stored and waiting to be used later."

 

Potential energy is due to the position of an object. It is not apparent until released. The stored energy of position is referred to as potential energy. Examples:

There are three types of potential energy:

 

 

1) Gravitational Potential Energy:

 

On earth, there is always have the force of gravity acting on us. When we separate from the earth's surface, we have the potential (stored) energy. This stored energy is called gravitational potential energy.

 

Consider a body of mass $m$ held stationary at height $h$ from the ground. The energy is released when the object falls to the ground, is equal to the work done by the body falling through this height. i.e.

 

P.E. = Work done by the force (or, Work done) in distance $h$.

or, P.E. = F * h

∴ P.E. = $m\,g\,h$ .......... (i) 

 

Where, $m$ = mass, $h$ = height, $g$ = gravitational field strength ($9.8\; N/kg$ on Earth).

 

The amount of gravitational potential energy an object on earth has depends on its Mass ($M$) and height above the ground ($h$).

2) Elastic potential energy:

 

Potential energy due to compression or expansion of an elastic objects. It can be found in rubber band and springs.

For certain springs (those obeys the HOOK'S Law), the amount of force is directly proportional to the amount of stretch or compression ($x$) is,

F$_{spring} = k . x$ .......... (i)

 

Where, $k$ is proportionality constant called spring constant.

In terms of potential energy, the special equation for springs that relates the amount of elastic potential energy to the amount of stretch (or compression) and the spring constant (k). The equation is,

P.E = $\frac{1}{2}k\,x^2$ .......... (ii)

 

Where, $k$ = spring constant
            $x$ = amount of compression or stretch (relative to equilibrium position).

 3) Chemical potential energy:

 

Potential energy stored with in a chemical bonds of an object. It is released when chemical reactions take place.

 

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E1.5 Work, Energy and Power (Kinetic Energy):

Energy is a central concept in Physics. It can be defined as the capacity for doing work. But, it is important to understand that just because energy exists, not necessarily available to do work.

Energy can not be create or destroyed, but it can change from one form to another form.

There are many types of energy such as: Mechanical energy, chemical energy, light energy, thermal energy, electrical energy etc. Mechanical energy further divided into two parts called: Kinetic Energy and Potential Energy. Here, we mainly focused on these two energy.

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Kinetic Energy: 

Kinetic energy is the energy of motion. It is a capacity of doing work due to its motion. There are many forms of kinetic energy - vibrational (energy due to vibrational motion), rotational (energy due to motion in circle) and translation (energy due to motion in straight line). Here, we will focus upon translation kinetic energy to keep matters simple.

In classical mechanics, the kinetic energy of an (translation) object has depends upon mass (m) and velocity (v) of an object. i.e.

$K.E. = \frac{1}{2}mv^2$

In relativistic mechanics, this is a good approximation only when $v$ is much less than the speed of light.

The amount of kinetic energy an object has depends on the mass and speed of the object.

All moving object have the kinetic energy. The faster object has more speed and has more kinetic energy. 

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» Calculation of Kinetic Energy:

Consider a body of mass $m$ lying on a horizontal surface. After applying a force $F$, the body travel a distance $d$ with a moving velocity $v$.

As we know from the definition of work:

W = force * distance = $F * d$ .......... (i)

From the Newton's law of motion:

$F = ma$ .......... (ii)

And from the Kinematics, we know the formula:

$v^2 = u^2 + 2ad$ .......... (iii)

If the initial velocity of a body is at rest, so the initial velocity is zero ($u = 0$), then equation (iii) becomes,

$ad = \frac{v^2}{2}$ .......... (iv)

From equation (i), (ii) and (iii), we have

$W = F * d = ma *d $

 $= m * ad = m * \frac{v^2}{2}$

Thus, 

∴ $W = K.E. = \frac{1}{2}mv^2$ .......... (v)

Where, m - mass and v = velocity of  a body.

This equation reveals that the kinetic energy of an object is directly proportional to the square of its speed. That means, for a two fold increase in speed, the kinetic energy will increase by the factor four and so on.

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