How Does Rotational Kinetic Energy Impact Everyday Objects?
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.
FAQs on Rotational Kinetic Energy Explained: Formula, Examples & Uses
1. What is rotational kinetic energy in simple terms?
Rotational kinetic energy is the energy an object possesses because of its rotation around an axis. Just as an object moving in a straight line has translational kinetic energy, an object spinning has rotational kinetic energy. It depends not only on how fast the object is spinning (its angular velocity) but also on its mass and how that mass is distributed relative to the axis of rotation (its moment of inertia).
2. What is the formula for calculating rotational kinetic energy?
The formula for rotational kinetic energy (K_rot) is K_rot = ½ Iω². In this equation:
- I represents the moment of inertia, which is a measure of an object's resistance to rotational motion.
- ω (omega) represents the angular velocity, which is how fast the object is rotating, measured in radians per second.
3. How does rotational kinetic energy differ from translational kinetic energy?
The primary difference lies in the type of motion. Translational kinetic energy (½mv²) is associated with an object's motion from one point to another in a line. In contrast, rotational kinetic energy (½Iω²) is associated with an object's spinning or rotating motion around an axis. A rolling wheel is a perfect example of an object possessing both, as it moves forward (translation) while also spinning (rotation).
4. What are some real-world examples of rotational kinetic energy?
Rotational kinetic energy is present in many everyday objects and systems. Some common examples include:
- The spinning blades of a ceiling fan or a wind turbine.
- A flywheel used in engines to store energy between power strokes.
- The Earth rotating on its axis, which contributes to its total energy.
- A spinning figure skater pulling their arms in to spin faster.
- A potter's wheel rotating as it shapes clay.
5. Why is moment of inertia used in the rotational kinetic energy formula instead of just mass?
While mass is the measure of inertia for linear motion, moment of inertia is the equivalent for rotational motion. It is crucial because in rotation, the distribution of mass is as important as the mass itself. An object with its mass concentrated far from the axis of rotation has a higher moment of inertia and more rotational kinetic energy than an object of the same mass and angular velocity with its mass concentrated near the axis. Therefore, the moment of inertia correctly accounts for both mass and its arrangement.
6. How do you calculate the total kinetic energy of a rolling object, like a ball or a disc?
A rolling object possesses both translational and rotational motion simultaneously. Therefore, its total kinetic energy is the sum of its translational and rotational kinetic energies. The formula is: K_total = K_translational + K_rotational, which expands to K_total = ½mv² + ½Iω². To solve this, you need the object's mass (m), linear velocity (v), moment of inertia (I), and angular velocity (ω).
7. How is the work-energy theorem applied to rotational motion?
The work-energy theorem for rotation states that the net work done by all external torques acting on a rigid body is equal to the change in its rotational kinetic energy. So, if a torque causes an object to speed up or slow down its rotation, the work done is the difference between its final and initial rotational kinetic energies (W = ΔK_rot = ½Iω_f² - ½Iω_i²). This is the rotational analogue of the work-energy theorem for linear motion.
8. What is the SI unit of rotational kinetic energy?
The SI unit for rotational kinetic energy is the same as for any other form of energy. It is the Joule (J). Despite involving different variables like moment of inertia and angular velocity, the final calculation results in a measure of energy, which is universally expressed in Joules in the SI system.
