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.

What 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 mv²

• 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 = 1/2Iω²

Where,

• K_R = rotational kinetic energy
• I = moment of inertia
• ω = angular velocity

The formula for rotational kinetic energy, K_R = 1/2Iω², 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 = 1/2Iω² for rotational motion and KE = 1/2mv² 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 = 1/2Iω²

Dimensional formula of moment of inertia = M¹L²T⁰

Dimensional formula of angular velocity = M⁰L⁰T^-1

Put the dimensional formula of moment of inertia and angular velocity in rotational kinetic energy, we get,

Dimensional formula of rotational kinetic energy = M¹L²T^-2

Unit of Rotational Kinetic Energy

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 m₁, m₂,…,m_n, located at distances r₁, r₂,…,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 = 1/2mv² , 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: E₁ = 1/2m₁r₁2ω2

Rotational KE of the second particle: E₂ = 1/2m₂r₂²ω²

Similarly, the rotational KE of the n^th particle is: E_n = 1/2m_nr_n²ω²

The total rotational kinetic energy of all the particles is the sum of their individual kinetic energies: E = E₁ + E2 +… + E_n

Substituting the expressions for E₁, E₂,…, E_n, we get:

E = 1/2(m₁r₁² + m₂r₂² +…+m_nr_n²)ω²

This expression can be further simplified using the concept of moment of inertia (I), defined as

I=∑_𝑖=1ⁿm_𝑖r_𝑖².

Thus, the equation becomes:

E=1/2Iω²

Therefore, the rotational kinetic energy (E) of an object with moment of inertia (I) and angular velocity (ω) is given by

E=1/2Iω²

Rotational Kinetic Energy of a Solid Sphere

To calculate the rotational kinetic energy of a solid sphere, we use the formula E =1/2Iω²

For a solid sphere rotating about its axis, the moment of inertia is given by I = 2/5mR²

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(2/5mR²)ω²

Simplifying this expression:

K=1/5mR²ω²

Therefore, the rotational kinetic energy of a solid sphere is:K = 1/5mRω²

Difference between Rotational and Translational Kinetic Energy

The difference between translational and rotational kinetic energy is given below:

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/m², 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 = 1/2Iω²

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 = 1/2Iω²

For a solid disc, the moment of inertia (I) is given by:

I = 1/2mr²

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 = 250π² Joules.

Example 3: An object with a moment of inertia of 2kg⋅m² is rotating at a constant angular speed of 4 rad/s. Calculate its rotational kinetic energy.

Solution:

Given:

Moment of inertia, I = 2 kg⋅m²

Angular velocity, ω = 4rad/s

We use the formula for rotational kinetic energy:

E = 1/2Iω²

Substitute the given values into the formula:

E = 1/2×2×(4)²

E = 1/2×2×16

E = 16 Joules