Rotational Kinetic Energy is described as the kinetic energy associated with the rotation of an object around an axis. It is also known as angular kinetic energy. It is dependent on the mass of an object and its angular velocity.
In this article, we will learn about rotational kinetic energy, its formula and derivation, examples of rotational kinetic energy, and the difference between rotational and translational kinetic energy.
What is Kinetic Energy?
Kinetic Energy is a fundamental concept in physics that is described as the energy of an object possessed due to the virtue of its motion. It is a scalar quantity. It is generally associated with the linear motion of an object. It depends on both the mass of the object and its velocity. The formula of kinetic energy is as follow:
KE = 1/2mv2
where,
- KE is the kinetic energy,
- m is the mass of the object, and
- v is its velocity.
Rotational Kinetic Energy
Rotational kinetic energy represents the energy associated with the rotational motion of an object. The magnitude of rotational kinetic energy depends on the mass of the object, the distribution of that mass around the axis of rotation (moment of inertia, I), and the angular velocity (ω).
Kinetic energy of an object possessed by the virtue of its rotational motion is known as Rotational kinetic energy. It is also known as angular kinetic energy.
Note: A rolling object possesses both rotational and translational kinetic energy.
Examples of Rotational Kinetic Energy
Some Examples that illustrate Rotational Kinetic Energy are:
- Rotation of Earth around its axis
- Rotating Wind Turbine Blades
Rotational Kinetic Energy describes the energy posed due to the rotational motion of an object around an axis. The Formula to calculate rotational kinetic energy of an object is given below:
KR = 1/2Iω2
Where,
- KR is the Rotational Kinetic energy,
- I is the moment of inertia,
- ω is the angular velocity.
Rotational kinetic energy is directly proportional to moment of inertia and the square of the angular velocity. This means that if either the moment of inertia or the square of the angular velocity increases, the rotational kinetic energy will also increase accordingly.
Also Check, Rotational Kinetic Energy Formula
Unit of Rotational Kinetic Energy
Unit of Rotational Kinetic Energy in different systems is given below:
System
| Unit
|
|---|
SI Unit
| Joules (J)
|
|---|
MKS
| kgm2s−2
|
|---|
CGS
| erg
|
|---|
Dimensional Formula of Rotational Kinetic Energy
Dimensional formula of rotational kinetic energy is M1L2T-2.
Rotational Kinetic Energy Derivation
To derive the formula of rotational kinetic energy consider the kinetic energy of individual particles within a rotating object, Let us assume that:
- m1, m2, …, mn are the masses of n particles in a rigid body rotating about an axis.
- r1, r2 ,…, rn are their distances from the axis of rotation respectively with angular velocity ω.
We know that,
The kinetic energy of an object is given by, K.E.=1/2mv2
And the relation between linear velocity and angular velocity is, v = rω
Therefore, rotational kinetic energy becomes, K.E. = 1/2mr2ω2
The rotational kinetic energy (K.E.) of each particle can be written as:
- Rotational K.E. of the first particle, E1 = 1/2m1r12ω2
- Rotational K.E. of the second particle. E2 = 1/2m2r22ω2
- Similarly, Rotational K.E. of nth particle will be En = 1/2mnrn2ω2
Hence the total Rotational kinetic energy of all the particles becomes:
E = E1 + E2 + ….. + En
Substitute the value of E1 , E2 ,…, En we get:
E = [1/2m1r12ω2 + 1/2m2r22ω2 + … + 1/2mnrn2ω2]
On simplifying it we get:
E = 1/2[m1r12+m2r22+…+mnrn2]ω2
E = 1/2 [[Tex]\sum_{i = 1}^{n}[/Tex]miri2] ω2
We know, the moment of inertia I = [Tex]\sum_{i = 1}^{n}[/Tex]miri2, Hence our equation becomes:
E = 1/2 I⋅ω2
Therefore, the rotational kinetic energy of an object with moment of inertia (I) and angular velocity (ω) is given by E = 1/2 I⋅ω2
Rotational Kinetic Energy of Solid Sphere
By using the kinetic energy formula, E = 1/2 Iω2 rotational kinetic energy of solid sphere can be calculated.
Solid sphere rotating about its axis has moment of inertia is, I = 2/5mR2
where m and R are mass and radius of solid sphere.
Therefore, by substituting value of I in the formula of rotational kinetic energy we get:
K = 1/2(2/5mR2)ω2
K = 1/5mR2ω2
Hence, the rotational Kinetic Energy of Solid Sphere is K = 1/5mR2ω2
Rotational Kinetic Energy of Earth
By using Rotational Kinetic Energy formula, E=1/2 Iω2 we can be calculate, rotational kinetic energy of Earth.
Earth’s moment of inertia is 8 × 1037 kgm²(approx.).
Its average angular velocity of Earth is 7.27×10-5 radians per second(approx.)
Putting these value in the formula of rotational kinetic energy we get:
E= 1/2 × 8×1037 kg m² × (7.27×10-5 rad/s)2
E= 1/2 × 8×1037 kg m² × 5.29 × 10 −9 kg m2 /s2
K = 2.12×1029 Joules
Hence, Earth’s rotational kinetic energy is approximately equal to 2.12×1029 Joules.
Rotational Kinetic Energy of Disc
By using the kinetic energy formula, E = 1/2 Iω2 rotational kinetic energy of disc can be calculated.
Disc rotating about its axis, has moment of inertia, I = 1/2mR2
where m and R are mass and radius of disc.
Therefore, by substituting the value of I we got the expression as,
K = 1/2(1/2mR2)ω2
K = 1/4mR2ω2
Hence, the rotational Kinetic Energy of a Disc is K = 1/4mR2ω2
Translational and Rotational Kinetic Energy
The difference between translational and rotational kinetic energy is illustrated in the table below:
Aspect
| Translational Kinetic Energy
| Rotational Kinetic Energy
|
|---|
Definition
| Energy due to linear motion of an object.
| Energy due to rotational motion of an object around an axis.
|
|---|
Formula
| Ktranslational = 1/2mv2
| Krotational = 1/2Iω2
|
|---|
Motion Type
| Linear motion in a straight line.
| Circular or rotational motion around an axis.
|
|---|
Example
| A car moving along a road.
| Earth rotating on its axis.
|
|---|
Solved Examples on Rotational Kinetic Energy
Example 1. A flywheel is spinning at an angular velocity of 100 rad/s. If its moment of inertia is 0.5 kgm2 , what is its rotational kinetic energy?
Solution:
Given, I = 0.5 kgm2 and ω = 100 rad/s
Use the formula for rotational kinetic energy: KE = 1/2Iω2
KE= 1/2 × 0.5 × (100)2 = 2500 Joules.
Example 2. A bicycle wheel of mass 10 kg has a radius of 0.5 meters and is spinning at 10 revolutions per second. Calculate its rotational kinetic energy.
Solution:
First, convert revolutions per second to radians per second:
Angular velocity = 10 × 2π = 20π rad/s
Use formula for rotational kinetic energy: KE = 1/2Iω2
The moment of inertia for a solid disc is I= 1/2mr2 ,where m is the mass and r is the radius.
Therefore,
I = 1/2 × 10 × (0.5)2 = 1.25
KE= 1/2 × 1.25 × (20π)2 = 250π2 Joules.
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Frequently Asked Questions (FAQs) on Rotational Kinetic Energy
What is the rotational kinetic energy of a ball?
The rotational kinetic energy of a ball (sphere) is given by K = 1/5mR2ω2
What is the formula of rotational kinetic energy?
The formula of rotational kinetic energy is KR = 1/2Iω2
What type of kinetic energy does a rolling object possess?
A rolling object possess both translational and rotational kinetic energy.
Explain Kinetic Energy.
The energy of an object possessed due to the virtue of its motion is known as its kinetic energy.
What is the dimensional formula of rotational kinetic energy?
The dimensional formula of rotational kinetic energy is M1L2T-2.
What is the SI unit of rotational kinetic energy?
The SI unit of rotational kinetic energy is Joules (J).
What is the statement of Work-Energy Theorem?
The work done on an object is equal to the change in its kinetic energy.
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