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Rotational Energy

When we see a giant wheel we would be excited to see it rotating. We even enjoy sitting in it. So, if we start thinking about the physics behind giant wheel we will know that it acquires both kinetic and potential energy. Since it is moving, it possesses Kinetic energy and if it attains some height it possesses potential energy. Hence we can say that rotational energy possesses both kinetic and potential energy. We will discuss about rotational energy in this section and also about how much kinetic and potential energy it possesses.

Rotational Energy Example

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Rotational Kinetic Energy

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Kinetic energy is of two types:
  • Rotational Kinetic Energy
  • Translational Energy
As the name suggests Translational energy is the kinetic energy that is gained by an object during its translational motion or motion along a line in particular axis.
It is given by,
K.E = $\frac{1}{2}$ mv2 
m = mass of the object or body and
v = the velocity of the object or body

Similarly, rotational kinetic energy is the energy which is gained by an object or body on virtue of its rotation.
It is given by,
Ek = $\frac{1}{2}$ I $\omega^{2}$
where, 
I = moment of Inertia
$\omega$ = angular Velocity of the rotating body

Rotational Kinetic Energy Units

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The units of rotational kinetic energy = Joules for S.I. System

Let us derive the units by using the formula:
We know that, 
K.E (rotational) = $\frac{1}{2}$ I $\omega^{2}$
Unit of Inertia: I = m r2
Hence the unit for inertia is I = Kg m2

Also for angular momentum $\omega$ has unit as rad/s
So, K.E (rotational) = Kg m2 rad2/s2
= watt sec
= Joules

Rotational Kinetic Energy Formula

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The Mathematical formula for rotational K.E is:
As we know for translation K.E,
K.E = $\frac{1}{2}$ mv2 
where, 
m = mass
V = velocity of the body

Similarly, if we consider the rotational component then Rotational Kinetic energy is:
K.E = $\frac{1}{2}$I $\omega^{2}$
where, 
I = moment of inertia
$\omega$ = Angular Velocity

This is the formula for Rotational Kinetic Energy.

To find the Total Rotational Kinetic Energy, we know that change in kinetic energy in translational motion is given by

$\Delta$ K.E = $\frac{1}{2}$ $\Delta$ m v 2 
$\Delta$ K.E = $\frac{1}{2}$ $\Delta$ m r2 $\omega^{2}$ ..............(a)

Also

K.E total = $\sum$ $\Delta$ K.E 
= $\sum$ $\frac{1}{2}$ $\Delta$ m r2 $\omega^{2}$
K.E total= $\omega^{2}$ $\sum$ $\frac{1}{2}$ $\Delta$ m r2
$\frac{1}{2}$ I $\omega^{2}$..............(b)


Here, I is the moment of inertia about the given axis of rotation
ETotal = E Translational + E Rotational
ETotal = $\frac{1}{2}$ $\Delta$ m v2 + $\frac{1}{2}$ I $\omega^{2}$ ......(c)
which gives the formula for Total Rotational Kinetic Energy.

Rotational Potential Energy

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The Potential energy gained by an object due to the virtue of its rotation is called rotational potential energy.As we know for translational motion the potential energy is the energy stored.
Potential energy is given by

U = - $\int$ F. d s ....................(a)
where, 
F = Force and
s = displacement

F(x) = - $\frac{dU}{dx}$ ................(b)
where, 
dU = Change in potential energy
dx = Small area

Similarly, for rotational motion, we have Potential energy as,

U = - $\int$ T. d$\theta$ ........................(c)

and

T ($\theta$) = $\frac{dU}{d \theta}$ ..................(d)
T ($\theta$) = change in the potential energy with respect to small angular displacement $\theta$.

Conservation of Rotational Energy

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The law of Conservation of rotational energy states that the energy of a system remains constant.
or
Energy can neither be destroyed nor be created. It changes from one state to another. Hence for Rotational energy,
E = K.E + P.EOr
m g h = $\frac{1}{2}$ m v2 + $\frac{1}{2}$ m $\omega^{2}$.
where, m= mass of the body,
g = gravity,
h = height,
V = velocity of the body,
$\omega$ = Angular Velocity.

Rotational Kinetic Energy of the Earth

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Rotational Kinetic Energy of Earth

The rotational energy of the earth is an evergreen source of energy. It is an
inexhaustible resource. It will not cause any pollution, warming etc. The energy provided is 60,000,000,000 times the total electric usage of all Americans for one year.
Earth has inertia of rotation = 8.070 $\times$ 1037 kgm2,
I = $\frac{2}{5}$ m r2
m = 6 $\times$ 1024 Kg and
r = 6.4 $\times$ 106 m
It has an angular velocity of $\frac{6.2832}{86164.09}$ radians/sec approximately. It rotates once in a day.
Hence,
Rotational kinetic energy = $\frac{1}{2}$ I $\omega^{2}$
Rotational kinetic energy = 2.138 $\times$ 1029 J

Kinetic Energy of Rotating  Body

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For a Rotating body we will use the formula:

K.E rotational = $\frac{1}{2}$ I $\omega^{2}$.

Now, for a body the moment of inertia is given by 

I = $\frac{1}{2}$ m r2

Where, m = mass of the body
r = radius of the body

Hence,
K.E = $\frac{1}{2}$ I $\omega^{2}$
K.E = $\frac{1}{2}$ ($\frac{1}{2}$ m r2) $\omega^{2}$ ........... (a)

Also 1 revolution = 2 $\pi$ rad
Hence,
$\omega$ = 2 $\pi$ $\frac{n}{t}$ ...............(b)

where, n= no of rotations,
t = time taken.

So,
K.Erotational = $\frac{1}{2}$ ($\frac{1}{2}$ mr2) $2 \pi$ $(\frac{n}{t})^{2}$  
K.Erotational = $\frac{1}{4}$ mr2 ${2 \pi (\frac{n}{t})^{2}}$.
which gives Kinetic Energy of the rotating body.
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