Rotational Kinetic Energy

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Introduction to Rotational Kinetic Energy

A rigid body possesses two kinds of energy: kinetic energy and potential energy. A rigid body's potential energy is the energy stored up in the body due to its position and other stresses on the body. The kinetic energy of a rigid body is a form of energy possessed by a moving body by means of its motion. If work is done on an object by applying a net force, the object gains speed which in turn increases its kinetic energy. The kinetic energy of a body in motion is dependent on its mass and speed. This article will cover kinetic energy in rotational motion and learn about the formula for rotational energy.


Rotational Kinetic Energy

When an object spins about an axis, it possesses rotational kinetic energy. The kinetic energy of a rotating body is analogous to the linear kinetic energy and depends on the following factors:

  • The speed at which the object is rotating, the faster the speed more is the energy.

  • The angular kinetic energy is directly proportional to the mass of the rotating object.

  • The position of the point mass from the axis of rotation also determines its energy. The particles that are further from the rotation axis possess more rotational kinetic energy than the ones closer to the rotational axis.


The Formula for Rotational Energy

To calculate the kinetic energy of rolling motion, let us first consider the linear kinetic energy of a body with mass m which is moving with a velocity of v.

Linear kinetic energy = ½ * m * v2

This straightforward formula holds for all objects moving in a straight line and applies to the center of mass of the object. This way the object is approximated to a point mass.

Now, to describe the formula for Rotational Energy of a rotating object we need to describe the mass distribution of the object along the axis of rotation, denoting it by the moment of inertia kinetic energy, I. Moment of inertia is the measure of difficulty in changing the rotational motion of a body around the axis of rotation. The kinetic energy moment of inertia depends on the mass of the body and the distribution of the mass around the axis of rotation. 

I = m*r2, where I is the moment of inertia, m is the mass of the body and r is the distance of point mass from the rotational axis. The unit of moment of inertia is Kg.m2.


The Formula for Rotational Energy is Given by 

K.Erotational = ½ (I * ω2). Where I is the moment of inertia and ω is the angular velocity of the object expressed in radians per second. The unit of rotational kinetic energy is Joule.


The Analogy between Rotational and Translational Kinetic Energies

The formulas expressed for the rotational and translational kinetic energy are analogous to each other in the sense:

  • Mass in translational energy (m)  ≅ moment of Inertia in rotational kinetic energy (I)

  • Linear velocity in translational motion (v)  ≅ angular velocity in rotational motion (ω2)

If an object is rotating as well as it’s center of mass is moving in a straight line then the total kinetic energy is given by the sum of rotational and translational kinetic energies.

K.E.total = K.E.rotational + K.E.linear = ½ (I * ω2) + (½ * m * v2)


Examples and Application of the Rotational Kinetic Energy Formula

The formula for Rotational Energy has many applications and can be used to:

  • Calculate the simple kinetic energy of an object which is spinning.

  • Calculate the kinetic energy of an object that is rolling, i.e. there are rotational and translational kinetic energies involved.

Let us look at the example of the movement of Earth about its axis. Earth is spinning on its axis at an approximate rate of once in 24 hours. How do you calculate its rotational kinetic energy, assuming it has uniform density?

Solution

Earth’s radius = 6.37 × 106 m

Mass of Earth = 5.97 × 1024 kg

We need to first find the moment of inertia to calculate rotational kinetic energy. Considering the shape of the Earth as sphere we get:

Moment of inertia I of Earth = ⅖ * m * r2 = ⅖ * (5.97 × 1024 kg * (6.37 × 106 m)2) = 9.69 * 1037 Kg.m2

Angular velocity of Earth = 2π radians in a day, which is converted into rad/s as:

2π/86400 seconds = 7.27 * 10-5 rad/sec

The rotational Kinetic energy of Earth = ½ (I * ω2)  = ½ * (9.69 * 1037 Kg.m2) * (7.27 * 10-5 rad/sec) = 2.56 * 1029 Joules


Newton’s Second Law of Rotation

To explain Newton’s 2nd law of rotation, let us first understand a few terms related to the theorem:

  • Torque - The twisting or rotational effect of a force on an object is called torque. It is expressed in newton-meter (Nm). The symbol for torque is τ.

  • Angular Acceleration - This is the change in the angular velocity of a rotating object per unit of time. The symbol for angular acceleration is 𝞪.

Newton’s 2nd law of rotation states that if several torques are acting on a rigid body about a fixed axis, then the total of the torques is equal to the moment of inertia times angular acceleration. In formula it is expressed as:

iτi = I * 𝞪

Here I𝞪 is a scalar quantity that can either be positive or negative depending on whether the rotation is clockwise or counterclockwise. Counterclockwise angular acceleration is considered positive, and clockwise is considered negative.

FAQ (Frequently Asked Questions)

1. What is the work-energy principle?

According to the work-energy theorem, the change in the kinetic energy of an object equals the net work done by forces on it. It is expressed as:

Wnet = K - K0 = ΔK, where Wnet is the net work done, K is the final Kinetic energy and K0 is the initial Kinetic energy. The net work is calculated by subtracting the initial kinetic energy of the object from its final kinetic energy.


Wnet =( ½ m * v2final) - (½ * m * v2initial)


This work-energy formula is used widely in solving mechanical problems and it can be derived from the law of conservation of energy.

2. What is the law of conservation of energy?

Energy is defined as the capacity to do work. The energy in the universe exists in many forms, and energy can be transformed from one form to another. But the energy transformations have a constraint that is based on the principle of conservation of energy. This principle states that energy can not be created and can not be destroyed. In other words, the total energy of an isolated system remains unchanged. An isolated system is defined as a collection of matter that does not interact with the rest of the universe in any manner.

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