Name: rotational inertia and energy
Uploaded: 2023-04-30T11:00:55.000Z
Duration: 1 min 14 s
Channel: JoVE

K12 Standards

Tennessee Academic Standards

High School

Physics

motion-and-stability-forces-and-interactions

Motion and Stability: Forces and Interactions

PHYS.PS2.3

Moment of Inertia

When a rigid body is in linear motion, every point moves with the same velocity. However, when it is in rotational motion, different points on the body have different velocities and therefore different kinetic energies.
If the i-th particle, placed at a perpendicular distance ri from the axis of rotation, has a tangential velocity of vi, its kinetic energy is calculated by replacing vi with the product of angular speed and ri.
The total kinetic energy of the rigid body is the sum of the individual kinetic energies of the constituent particles. Since angular speed is the same for each particle, it can be pulled out of the summation.
On comparing this equation with the kinetic energy equation for translational motion, a new rotational variable is defined. This quantity is called the moment of inertia and has units of kilogram meters squared.
For a single particle rotating about a fixed axis, the moment of inertia is the product of its mass and the square of its distance from the axis of rotation.

moment of inertia

The comparability between linear and angular velocities, linear and angular accelerations, and the kinematic equations of translational and rotational motion can be extended to the concept of inertia.
If a rigid body is rotating about an axis but is not in translational motion, its translational kinetic energy is zero. However, since each particle undergoes rotational motion, it possesses non-zero velocity and kinetic energy. Thus, the kinetic energy of the rigid body, which is the sum of the kinetic energy of its constituents, is non-zero. The rotational kinetic energy of a rigid body is given as half the square of the angular speed times the moment of inertia.
Rigid bodies and systems of particles with more mass concentrated at a greater distance from the axis of rotation have greater moments of inertia than bodies and systems of the same mass but concentrated near the axis of rotation. For instance, a hollow cylinder has more rotational inertia than a solid cylinder of the same mass when rotating about an axis through the center.
Although defined while keeping rigid bodies in mind, the moment of inertia also applies to single particles. It helps treat&#160;objects rotating with respect to inertial frames of reference as single particles with their total mass concentrated at the center of mass.
This text is adapted from <a href="https://openstax.org/books/university-physics-volume-1/pages/10-4-moment-of-inertia-and-rotational-kinetic-energy">Openstax, University Physics Volume 1, Section 10.4: Moment of Inertia and Rotational Kinetic Energy.</a>

The comparability between linear and angular velocities, linear and angular accelerations, and the kinematic equations of translational and rotational ...

Education

JoVE Core

Unit 1

Rotation and Rigid Bodies

KEY TERMS AND CONCEPTS

rotational inertia and energy

Rotational inertia and rotational kinetic energy describe how objects resist changes in rotation. The idea connects angular speed, angular acceleration, and the kinematic equations for rotational motion to the same inertia concept used in linear motion.

A rigid body that is rotating about an axis, but not moving in translation, has zero translational kinetic energy. Even so, each particle in the body still moves in a circle and has velocity. Because of that motion, the body has rotational kinetic energy, which equals one-half the square of the angular speed times the moment of inertia.

Moment of inertia depends on how mass is spread out from the axis of rotation. When more mass is farther from the axis, the moment of inertia is larger. A hollow cylinder therefore has more rotational inertia than a solid cylinder with the same mass when both rotate about an axis through the center.

The moment of inertia also applies to a single particle. In rotational problems, it helps treat objects in inertial frames of reference as if their total mass were concentrated at the center of mass.

Rotational Inertia and Energy

angular velocity and displacement

Rotational Motion in a Circle

Rotational motion in a circle uses angular position, angular displacement, and angular velocity to describe motion around a fixed center. Uniform circular motion is the simplest case because the object moves in a circle at a constant speed. It is still useful for many situations and helps introduce rotational variables. A fixed coordinate system serves as the frame of reference for the particle's position. The position vector from the origin of the circle to the particle sweeps out an angle,...

angular velocity and acceleration

Angular Acceleration in Rotational Motion

Angular acceleration describes how quickly angular velocity changes in rotational motion. It matters when motion is not uniform. An ice skater spinning with arms outstretched can increase angular velocity by pulling the arms inward. A computer hard disk can also slow down, which means its angular velocity decreases. The faster angular velocity changes, the greater the angular acceleration. Instantaneous angular acceleration is defined as the derivative of angular velocity with respect to time.

rotation with constant angular acceleration - i

rotation-with-constant-angular-acceleration-i

Rotational Motion Under Constant Acceleration

Rotational motion under constant angular acceleration uses a simple set of kinematic equations. These equations work much like the equations for linear motion, but they describe spinning objects instead of objects moving in a straight line. When angular acceleration stays constant, several rotational quantities are linked in a predictable way. Angular displacement, angular velocity, angular acceleration, and time all change together. If the angular acceleration points in the same direction as...

rotation with constant angular acceleration - ii

rotation-with-constant-angular-acceleration-ii

Rotational Kinematics Using Fixed-Axis Motion

Rotational kinematics describes motion in terms of rotation angle, angular velocity, angular acceleration, and time. It shows how these quantities are related for an object turning about a fixed axis. Kinematics is a way to describe motion, not explain its cause. In rotational motion, angular acceleration means a rapid change in angular velocity. A large angular acceleration tells us how quickly the rotation rate changes. It does not tell us what causes that change. Because of this, rotational...

relating angular and linear quantities - i

Angular and Linear Motion Connections

Angular and linear motion use matching ideas to describe movement, especially in circular motion. This page links position, velocity, and acceleration in straight-line motion to their rotational counterparts. It also shows how these quantities are measured and how they change for a particle moving in a circle. Linear position is measured in meters, while angular position is measured in radians. Radians are dimensionless because they are a ratio of two lengths. Linear velocity uses units of...

relating angular and linear quantities - ii

relating-angular-and-linear-quantities-ii

Centripetal and Tangential Acceleration

Centripetal and tangential acceleration describe motion in circular paths. For a particle moving at a radius from a rotating axis, linear tangential speed is related to angular velocity. The same idea can also apply to points on a rigid body rotating about a fixed axis. Circular motion, whether uniform or nonuniform, includes centripetal acceleration. Centripetal acceleration is the inward acceleration that points from the particle toward the axis of rotation. In uniform circular motion,...

Rotational inertia and rotational kinetic energy describe how objects resist changes in rotation. The idea connects angular speed, angular acceleration, and the kinematic equations for rotational motion to the same inertia concept used in linear motion. A rigid body that is rotating about an axis, but not moving in translation, has zero translational kinetic energy. Even so, each particle in the body still moves in a circle and has velocity. Because of that motion, the body has rotational...

moment of inertia and rotational kinetic energy

moment-of-inertia-and-rotational-kinetic-energy

Rotational Inertia and Energy Storage

Rotational kinetic energy depends on both angular speed and moment of inertia. The moment of inertia is the rotational version of mass, so it describes how hard it is to spin an object. For the same angular speed, an object with a larger moment of inertia has more rotational kinetic energy. Because of this, more work is needed to change the rotational kinetic energy and bring the object to a chosen angular speed. The moment of inertia therefore measures rotational inertia, just as mass...

moment of inertia: calculations

Calculating Rotation Resistance

Moment of inertia measures how hard it is to rotate an object, and the answer depends on the axis of rotation. A simple barbell shows this clearly. It has two masses at the ends of a rod. The masses can be treated as point masses, and the rod can be assumed to have negligible weight. Once the rotation axis is chosen, the moment of inertia can be found from the mathematical definition. If the two point masses each have mass m and are a distance R from the center of the rod, the moment of...

moment of inertia of compound objects

Calculating Rotational Inertia in Combined Systems

Rotational inertia, also called moment of inertia, describes how hard it is to rotate an object. It depends on how mass is spread out from the axis of rotation. For compound objects, the total moment of inertia is found by adding the moments of inertia of the individual parts. A simple example uses a child on a rotating merry-go-round. The child has a mass of 25 kg and stands 1 m from the axis. The merry-go-round is modeled as a uniform solid disk with a mass of 500 kg and a radius of 2 m. To...

parallel-axis theorem

Moment of Inertia with Parallel Axes

The parallel-axis theorem helps find the moment of inertia of an object about a new axis that is parallel to the axis through its center of mass. It gives a fast way to handle rotation problems when the axis is shifted away from the center. A thin rod shows why this method is useful. Finding the moment of inertia of a rod about its middle uses the center of mass, while finding it about one end uses an axis at the edge. The second case can be done with the usual method by using linear mass...

perpendicular-axis theorem

Moment of Inertia for a Flat Disc

The moment of inertia of a flat disc can be found using the perpendicular-axis theorem. For a planar object, the moment of inertia about an axis perpendicular to its plane equals the sum of the moments of inertia about two mutually perpendicular axes in the plane of the body. For a circular disc with mass M and radius R, the disc lies in the x-y plane. The origin is at the center of the disc. The z-axis is perpendicular to the disc’s plane, and all three axes meet at the center. For this...

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vector transformation in rotating coordinate systems

vector-transformation-in-rotating-coordinate-systems

Velocity and Acceleration in Rotating Frames

Velocity and acceleration in rotating frames are found by comparing a vector measured in an inertial frame with the same vector measured in a rotating frame. A vector that turns with a fixed angular velocity traces a circular path. In a short time interval, it moves through an angle equal to angular velocity multiplied by time. The tip of the vector shifts by a change in magnitude called ΔB, and α is the angle between the vector and the angular velocity vector. For a vector in pure rotation...

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coriolis force

Earth’s Rotation and the Coriolis Effect

Earth’s rotation creates the Coriolis force, a pseudo force seen in rotating reference frames. It helps explain why motion looks different on a spinning surface than it does in an inertial frame, where Newton’s second law is written in its usual form. A sliding ball on a rotating table is one example of this change in motion. In a non-inertial frame, fictitious forces must be added to the motion of an object. Centripetal force and Coriolis force are the two pseudo forces that act on a particle...

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key terms and concepts

Explore how mass distribution affects rotational inertia, why rotating bodies still have kinetic energy, and why a hollow cylinder has greater inertia than a solid one.

Rotational Inertia and Energy | JoVE K12

physics

JoVE Core Physics introduces the theory and application of fundamental physics topics, including kinematics, dynamics, fluid mechanics, statics, waves, and acoustics, with concise and easy-to-understand animated videos. Additionally, the scientist-in-action videos demonstrate the application of related concepts in classical and original research experiments performed in today's laboratories worldwide.

Kinematic Equations - II

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PHYS.PS2.2

Angular Velocity and Displacement

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If the rotational definitions are compared with the definitions of linear kinematic variables from motion along a straight line and motion in two and ...

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This relationship between the ...

Algebraically solve problems involving arc length, angular velocity, and angular acceleration. Algebraically solve problems involving arc length, angular velocity, and angular acceleration. EXPLANATION: Though not explicitly stated, it is beneficial to develop this standard in the same manner which PHYS.PS2.1 is used to develop PHYS.PS2.2. In doing so, students can parallel rotational properties to translational properties, e.g., arc length can be seen as the rotational equivalent to displacement in the translational world. In doing so, radians becomes a logical unit of measure for rotational displacement. Since neither torque, nor moment of inertia are addressed in this course, discussions can be limited to considering only kinematics and not venturing into the realm of rotational dynamics.

Force

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Newton's Second Law

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Spherical Coordinates

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Work Done Over an Inclined Plane

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Projectile Motion

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Projectile Motion: Equations

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Projectile Motion: Example

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Use free-body diagrams to illustrate the contact and noncontact forces acting on an object. Use the diagrams in combination with graphic or component based analysis with Newton’s first and second laws to predict position of the object on which the forces act in a constant net force scenario. Use free-body diagrams to illustrate the contact and noncontact forces acting on an object. Use the diagrams in combination with graphic or component based analysis with Newton’s first and second laws to predict position of the object on which the forces act in a constant net force scenario. EXPLANATION: The concept of net force and Newton’s laws have been introduced in 8.PS2.3 and 8.PS2.4. At that time, free-body diagrams are introduced as a tool to represent the forces acting on an object. In further developing the use of free-body diagrams, it is now appropriate to include vectors that must be evaluated to parallel and perpendicular components. This includes objects on inclined planes as well as projectile motion as addressed in PHYS.PS2.13.

PHYS.PS2.4

Mass and Weight

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Gravitation

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Newton's Law of Gravitation

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Free-falling Bodies: Introduction

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Acceleration due to Gravity on Earth

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the-principle-of-superposition-and-the-gravitational-field

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Hooke's Law and Simple Harmonic Motion

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Frequency of Spring-Mass System

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Elastic Potential Energy

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Gather evidence to defend the claim of Newton’s first law of motion by explaining the effect that balanced forces have upon objects that are stationary or moving at a constant velocity.Gather evidence to defend the claim of Newton’s first law of motion by explaining the effect that balanced forces have upon objects that are stationary or moving at a constant velocity. EXPLANATION: Students should be able to discuss mass as a measurement of the amount of inertia in an object, with a unit of kg. Beginning as early as 5.PS2.3 students have been developing a understanding of gravity. Experiments determining the rate of acceleration of objects in freefall can be used to determine earth’s gravitational field strength which can then be used to develop an object’s gravitational mass. A discussion of Hook’s Law and subsequent use of a spring with a known spring constant can be used to determine an object’s inertial mass.

PHYS.PS2.5

Second Law: Motion under Same Force

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second-law-motion-under-same-acceleration

Second Law: Motion under Same Acceleration

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Linear Momentum

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Force and Momentum

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Impulse

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Impulse-Momentum Theorem

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Types Of Collisions - I

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Using experimental evidence and investigations, determine that Newton’s second law of motion defines force as a change in momentum, F = ∆p/∆t. Using experimental evidence and investigations, determine that Newton’s second law of motion defines force as a change in momentum, F = ∆p/∆t. EXPLANATION:  Previous examinations of Newton’s second law have been limited to instances with constant forces. This standard expands that discussion to include instances where the objects interact with each other. To maximize the quality of experimental investigations, magnets might be used to create situations where objects “collide” in an elastic manner. Newton’s second law can be expressed as F = ma. Viewing acceleration as a change in velocity over a period of time, one arrives at F=m(∆v/∆t). Distributing mass into this equation yields F = (mV-mVo)/∆t. A final recognition that momentum (p) is a property described by an object’s mass and velocity allows for substitution to produce F = ∆p/∆t.

PHYS.PS2.6

Rotational Inertia and Energy

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