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.

Moment of Inertia

The total of mr² for all the point masses that make up an object's moment of inertia I, where m is the mass and r is the distance of the mass from the centre of mass, may be described as the object's moment of inertia I. It may be mathematically stated as I = ∑mr2. Here, I is analogous to m in translational motion. In translational motion, I is comparable to m.

The axis around which you spin an object also affects the moment of inertia. Objects spin about their centre of mass by default, but they may be configured to rotate around any axis. The parallel axis theorem calculates the moment of inertia when rotating around an axis other than the centre of mass. The moment of inertia for an object rotated about a different axis parallel to the axis passing through the centre of mass isIcm+mr2, where r is the distance between the two axes and I cm is the moment of inertia when rotated about the centre of mass, which you learned how to calculate in the previous paragraph.

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 its 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 a 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 a 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.

Torque

The rotational equivalent of linear force is torque. Depending on the subject of study, it is also known as the moment, moment of force, rotating force, or turning effect. Archimedes' study of the use of levers gave birth to the idea. Torque is a twist of an item around a given axis, similar to how a linear force is a push or a pull. The product of the magnitude of the force and the perpendicular distance of the line of action of a force from the axis of rotation is another definition of torque.

Torque is usually represented by the lowercase Greek letter tau or symbol 𝛕. When the term "moment of force" is used, it is usually abbreviated as M.

Flywheel 

The usage of a flywheel can smooth out energy fluctuations and make the energy flow of an intermittently working machine more consistent. In most combustion piston engines, flywheels are employed.

In a flywheel, energy is stored mechanically as kinetic energy. A flywheel that stores and provides mechanical energy in the form of rotating kinetic energy has been studied for decades, but its use in automotive systems has been limited due to its hefty weight and expensive cost. The development of flywheel energy storage systems has been expedited by recent advancements in frictionless magnetic bearings, carbon-fiber composite materials, manufacturing techniques, and advanced power electronic controllers.

Conclusion

Rotational Kinetic Energy is a form of energy possessed by a moving body by means of its motion. 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.