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Rotational Kinetic Energy Formula
Rotational kinetic energy, also referred to as angular kinetic energy, is the energy linked with the rotation of an object around an axis. Unlike translational kinetic energy, which relates to the linear motion of an object, rotational kinetic energy pertains to its rotational motion. This type of energy relies on two main factors: the mass of the object and its angular velocity.In this article, we’ll explore the concept of rotational kinetic energy, its formula, how it’s derived, examples where it’s applicable.
Quick Links
ToggleWhat is Kinetic Energy Formula?
Kinetic energy is the energy an object possesses due to its motion. It arises when an object is accelerated by the application of forces, resulting in work being done on the object. This work transfers energy to the object, causing it to move at a constant velocity. The kinetic energy of an object depends entirely on its speed and mass.
For the Kinetic formula, k, is certainly the energy of a mass, m, motion, of course, is v2.
k = 1/2 mv^{2}
 k = Kinetic energy
 m = mass of the body
 v = velocity of the body
What is Rotational Kinetic Energy?
Rotational kinetic energy embodies the energy inherent in an object’s rotational motion. This form of energy is intricately linked to the mass of the object, the arrangement of that mass concerning the axis of rotation (quantified as the moment of inertia, I), and the angular velocity (ω). Termed as rotational kinetic energy, this aspect of kinetic energy pertains specifically to the rotational motion of objects. It’s also referred to as angular kinetic energy, underlining its association with the angular movement of the object.
Examples of Rotational Kinetic Energy
Some Examples related to Rotational Kinetic Energy are given below:
 Spinning Top
 Rotation of Earth around its axis
 Rotating Fan Blades
 Rotating Wheels of a Car
 Rotating Wind Turbine Blades
Rotational Kinetic Energy Formula
The formula of rotational kinetic energy is given as follows:
K_{R} = 1/2Iω^{2}
Where,
 K_{R} = rotational kinetic energy
 I = moment of inertia
 ω = angular velocity
The formula for rotational kinetic energy, K_{R} = 1/2Iω^{2}, illustrates the energy associated with the rotational motion of an object. It’s derived from the principle of work and energy, akin to the formula for linear kinetic energy. In a parallel scenario, imagine exerting a constant torque on a wheel with a moment of inertia (I) and applying a force to a mass (m), both initially at rest. The torque and force act to impart rotational and translational motion, respectively. As the wheel rotates and the mass moves, kinetic energy is generated. This energy can be calculated using the respective formulas:
K_{R} = 1/2Iω^{2} for rotational motion and KE = 1/2mv^{2} for linear motion.
Through this parallel, we see how both rotational and linear kinetic energies arise from the application of force or torque, resulting in motion from rest.
Rotational kinetic energy is directly proportional to the moment of inertia and the square of the angular velocity. In simpler terms, if either the moment of inertia or the square of the angular velocity experiences an increase, the rotational kinetic energy will also rise accordingly. This means that changes in either of these factors will directly impact the amount of energy associated with the rotational motion of an object.
Rotational Kinetic Energy Dimensional Formula
We know that rotational kinetic energy is given as: K_{R} = 1/2Iω^{2}
Dimensional formula of moment of inertia = M^{1}L^{2}T^{0}
Dimensional formula of angular velocity = M^{0}L^{0}T^{1}
Put the dimensional formula of moment of inertia and angular velocity in rotational kinetic energy, we get,
Dimensional formula of rotational kinetic energy = M^{1}L^{2}T^{2}
Unit of Rotational Kinetic Energy
System  Unit 
SI Unit  Joules (J) 
MKS  kgm2s−2 
CGS  erg 
Derivation of Rotational Kinetic Energy
To derive the formula for rotational kinetic energy, we consider the kinetic energy of individual particles within a rotating object. Let’s assume that the object consists of n particles with masses m_{1}, m_{2},…,m_{n}, located at distances r_{1}, r_{2},…,r_{n} from the axis of rotation and rotating with an angular velocity ω.
We know that the kinetic energy of an object is given by KE = 1/2mv^{2} , where v is the linear velocity. Since linear velocity (v) is related to angular velocity (ω) by v = rω, where r is the distance from the axis of rotation, the rotational kinetic energy (KE) becomes KE = 1/2mr2ω.
The rotational kinetic energy of each particle can thus be expressed as:
Rotational KE of the first particle: E_{1 }= 1/2m_{1}r_{1}2ω2
Rotational KE of the second particle: E_{2 }= 1/2m_{2}r_{2}^{2}ω^{2}
Similarly, the rotational KE of the n^{th} particle is: E_{n} = 1/2m_{n}r_{n}^{2}ω^{2}
The total rotational kinetic energy of all the particles is the sum of their individual kinetic energies: E = E_{1 }+ E2 +… + E_{n}
Substituting the expressions for E_{1}, E_{2},…, E_{n}, we get:
E = 1/2(m_{1}r_{1}^{2 }+ m_{2}r_{2}^{2 }+…+m_{n}r_{n}^{2})ω^{2}
This expression can be further simplified using the concept of moment of inertia (I), defined as
I=∑_{𝑖=1}^{n}m_{𝑖}r_{𝑖}^{2}.
Thus, the equation becomes:
E=1/2Iω^{2}
Therefore, the rotational kinetic energy (E) of an object with moment of inertia (I) and angular velocity (ω) is given by
E=1/2Iω^{2}
Rotational Kinetic Energy of a Solid Sphere
To calculate the rotational kinetic energy of a solid sphere, we use the formula E =1/2Iω^{2}
For a solid sphere rotating about its axis, the moment of inertia is given by I = 2/5mR^{2}
where, m is the mass of the sphere and R is its radius.
By substituting the value of I into the rotational kinetic energy formula, we get:
K=1/2(2/5mR^{2})ω^{2}
Simplifying this expression:
K=1/5mR^{2}ω^{2}
Therefore, the rotational kinetic energy of a solid sphere is:K = 1/5mRω^{2}
Difference between Rotational and Translational Kinetic Energy
The difference between translational and rotational kinetic energy is given below:
Rotational Kinetic Energy  Translational Kinetic Energy 
It is Defined as Energy due to rotational motion of an object around an axis.  It is Defined as Energy due to linear motion of an object. 
Kr = 1/2Iω^{2}  Kt= 1/2mv^{2} 
Circular or rotational motion around an axis.  Linear motion in a straight line. 
Example is Earth rotating on its axis.  Example is A car moving along a road. 
Solved Examples on Rotational Kinetic Energy
Example 1: A flywheel has an angular velocity of 100 rad/s. If its moment of inertia is 0.5 kg/m^{2}, calculate its rotational kinetic energy.
Solution:
Given:
Moment of inertia, I = 0.5 kg·m²
Angular velocity, ω = 100 rad/s
We use the formula for rotational kinetic energy:
KE = 1/2Iω^{2}
Substitute the given values into the formula:
KE = 1/2×0.5×(100)^{2}
KE = 1/2×0.5×10000
KE = 0.25×10000
KE = 2500 Joules
Example 2: A bicycle wheel has a mass of 10 kg and a radius of 0.5 meters. It is spinning at a rate of 10 revolutions per second. Determine its rotational kinetic energy.
Solution:
First, convert the angular velocity from revolutions per second to radians per second:
Angular velocity = 10 revolutions/second×2π radians/revolution = 20π rad/s
Next, use the formula for rotational kinetic energy:
KE = 1/2Iω^{2}
For a solid disc, the moment of inertia (I) is given by:
I = 1/2mr^{2}
where m is the mass and r is the radius.
Substitute the given values into the moment of inertia formula:
I = 1/2×10 kg×(0.5 m)^{2}
I = 1/2×10×0.25
I = 1.25
KE = 1/2 × 1.25 × (20π)^{2} = 250π^{2} Joules.
Example 3: An object with a moment of inertia of 2kg⋅m^{2} is rotating at a constant angular speed of 4 rad/s. Calculate its rotational kinetic energy.
Solution:
Given:
Moment of inertia, I = 2 kg⋅m^{2}
Angular velocity, ω = 4rad/s
We use the formula for rotational kinetic energy:
E = 1/2Iω^{2}
Substitute the given values into the formula:
E = 1/2×2×(4)^{2}
E = 1/2×2×16
E = 16 Joules
FAQs (Frequently Asked Questions)
1. What is rotational kinetic energy and how is it calculated?
Rotational kinetic energy is the energy that a rotating object possesses due to its angular motion. It can be calculated using the formula: 𝐾𝐸=1/2𝐼𝜔^{2}
2. How does the moment of inertia affect rotational kinetic energy?
The moment of inertia (I) measures how the mass of an object is distributed in relation to the axis of rotation. A higher moment of inertia indicates that more mass is positioned farther from the axis, which results in greater rotational kinetic energy for a specific angular velocity.
3. What are the units used for rotational kinetic energy?
In the International System of Units (SI), rotational kinetic energy is measured in joules (J), which is equivalent to kg⋅m^{2}/s^{2}
4. Can rotational kinetic energy be negative?
Rotational kinetic energy cannot be negative. It depends on the square of the angular velocity (𝜔^{2} ) and the moment of inertia (I), both of which are nonnegative, ensuring that the rotational kinetic energy is always zero or positive.
5. How is rotational kinetic energy different from translational kinetic energy?
Rotational kinetic energy relates to an object’s motion around a fixed axis and is dependent on its moment of inertia and angular velocity. In contrast, translational kinetic energy pertains to the straightline motion of an object and is calculated by 1/2𝑚𝑣^{2} , where m is the mass and v is the linear velocity.