Name: rotational motion under constant acceleration
Uploaded: 2023-04-30T11:00:25.000Z
Duration: 1 min 37 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

rotation-with-constant-angular-acceleration-i

Rotation with Constant Angular Acceleration - I

The equations derived for linear motion also hold true for rotational motion if the linear motion variables are replaced by their rotational motion counterparts.
Let &#969;0z be the angular velocity of the rotating object at any time t equal to zero and &#969;z be its final angular velocity at later time t. If the angular acceleration, &#945;z, of the object is constant, it can be written as the difference between final and initial angular velocity over time t.
Rearranging this expression, the first kinematic equation for rotational motion is obtained. Thus, the angular velocity at any time equals the sum of initial angular velocity and the change in angular velocity.
To derive the second equation, two equations for average angular velocity are used. One, average angular velocity equals the total change in angular displacement over time t.
Two, for constant acceleration, average angular velocity equals the average of the initial and final velocity. Solving these two equations, the second equation of rotational motion is obtained.
Here, the angular position of an object at any time is represented as the sum of its initial angular position, the displacement moved under constant initial angular velocity, and the angular displacement traveled during the change in angular velocity.

rotation with constant angular acceleration - i

If angular acceleration is constant, then we can simplify equations of rotational kinematics, similar to the equations of linear kinematics. This simplified set of equations can be used to describe many applications in physics and engineering where the angular acceleration of a system is constant.
Using our intuition, we can begin to see how rotational quantities such as angular displacement, angular velocity, angular acceleration, and time are related to one another. For example, if a flywheel has an angular acceleration in the same direction as its angular velocity vector, its angular velocity increases with time, as does its angular displacement. On the contrary, if the angular acceleration is opposite in direction to the angular velocity vector, its angular velocity decreases with time. These physical situations, along with many others, can be described with a consistent set of rotational kinematic equations under constant angular acceleration. The method to investigate rotational motion in this way is called kinematics of rotational motion.
To begin, note that if a system is rotating under constant acceleration, then the average angular velocity follows a simple relation, because the angular velocity is increasing linearly with time. The average angular velocity is simply half the sum of the initial and final values. From this, an equation relating the angular position, average angular velocity, and time can be obtained.
This text is adapted from <a href="https://openstax.org/books/university-physics-volume-1/pages/10-2-rotation-with-constant-angular-acceleration">Openstax, University Physics Volume 1, Section 10.2: Rotational </a><a href="https://openstax.org/books/university-physics-volume-1/pages/10-2-rotation-with-constant-angular-acceleration">with Constant Angular Acceleration.</a>

If angular acceleration is constant, then we can simplify equations of rotational kinematics, similar to the equations of linear kinematics. This ...

Education

JoVE Core

Unit 1

Rotation and Rigid Bodies

KEY TERMS AND CONCEPTS

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 the angular velocity vector, the angular velocity increases with time and the angular displacement also grows. If the angular acceleration points in the opposite direction, the angular velocity decreases with time.

These situations are common in physics and engineering, where a rotating system may speed up or slow down at a steady rate. A flywheel is one example used to picture this idea. The study of motion in this way is called kinematics of rotational motion.

A useful starting point is the average angular velocity. For constant angular acceleration, angular velocity changes linearly with time, so the average angular velocity is half the sum of the initial and final angular velocities. That relation leads to an equation that connects angular position, average angular velocity, and time.

Rotational Motion Under Constant Acceleration

rotational-motion-under-constant-acceleration

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.

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,...

moment of inertia

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...

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 constant angular acceleration links angular displacement, angular velocity, and time using rotational kinematics and a flywheel example.

Rotational Motion Under Constant Acceleration | 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.

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Algebraically solve problems including constant velocity and constant acceleration in onedimension. Algebraically solve problems including constant velocity and constant acceleration in onedimension. EXPLANATION:  Students can use the models they have developed in PHYS.PS2.1 to evaluate systems. For a system to undergo constant acceleration, the net force on the object must be constant throughout the problem. Algebraic problem solving should be extended to include proportional reasoning, beyond simple manipulation of variables.

PHYS.PS2.2

Angular Velocity and Displacement

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Rotation with Constant Angular Acceleration - II

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Relating Angular And Linear Quantities - II

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Moment of Inertia

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Moment of Inertia and Rotational Kinetic Energy

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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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vector-components-in-the-cartesian-coordinate-system

Vector Components in the Cartesian Coordinate System

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Polar and Cylindrical Coordinates

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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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Free-falling Bodies: Example

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Hooke's Law and Simple Harmonic Motion

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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

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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 Motion Under Constant Acceleration

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