Name: rotational kinematics using fixed-axis motion
Uploaded: 2025-05-22T09:05:40.000Z
Duration: 1 min 16 s
Channel: JoVE

K12 Standards

Tennessee Academic Standards

High School

Physics

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Motion and Stability: Forces and Interactions

PHYS.PS2.3

rotation-with-constant-angular-acceleration-ii

Rotation with Constant Angular Acceleration - II

The first and second equations of rotational motion with constant angular acceleration both have time as a variable.
However, the third equation is independent of time. To derive the third equation, begin by rearranging the first equation of rotational motion to obtain an expression for time. Then, substitute the value of time in the second equation of rotational motion.&#160;&#160;
Now, rearrange the &#952;0&#160;term and multiply both sides by 2&#945;z. Simplifying it further gives an expression for the final angular velocity in terms of initial angular velocity, angular acceleration, and the difference between the final and initial angular displacements. This is the third equation of rotational motion.
On the other hand, the fourth equation can be obtained by substituting the first equation for rotational motion into the second equation.&#160;
This equation gives the relationship between the final angular position of an object with respect to the initial angular position, the rotation with constant angular velocity, and the rotation due to a change in angular velocity.&#160;

rotation with constant angular acceleration - ii

Kinematics is the description of motion. The kinematics of rotational motion discusses the relationships between rotation angle, angular velocity, angular acceleration, and time. One can describe many things with great precision using kinematics, but kinematics does not consider causes. For example, a large angular acceleration describes a very rapid change in angular velocity without any consideration of its cause. Thus, rotational kinematics does not represent the laws of nature.
The first two kinematic equations for rotational motion involve time as a variable, but the third is independent of time. This equation expresses the angular velocity of an object with respect to its initial angular velocity and total angular displacement due to constant angular acceleration. The fourth kinematic equation represents the final angular displacement of an object as a function of the initial angular displacement, the displacement under constant angular velocity, and displacement under the constant angular acceleration.&#160;
These kinematics equations enable us to solve many problems in physics and engineering. The choice of kinematic equations in solving a problem depends on the variables present in the problem. Sometimes, it is necessary to use multiple equations to solve a problem. It is important to note that all the rotational kinematics equations are valid only when an object is rotating about a fixed axis with constant angular acceleration.
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>

Kinematics is the description of motion. The kinematics of rotational motion discusses the relationships between rotation angle, angular velocity, angular ...

Education

JoVE Core

Unit 1

Rotation and Rigid Bodies

KEY TERMS AND CONCEPTS

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 kinematics describes motion but does not represent the laws of nature.

The first two rotational kinematic equations include time as a variable. The third equation does not depend on time. It relates final angular velocity to initial angular velocity and total angular displacement when angular acceleration is constant.

The fourth kinematic equation gives final angular displacement. It combines the initial angular displacement, the displacement from constant angular velocity, and the displacement from constant angular acceleration. These equations are useful for solving many physics and engineering problems. The choice of equation depends on which variables are given, and some problems require more than one equation.

These rotational kinematics equations work only for motion about a fixed axis with constant angular acceleration. That condition is important when solving problems. If the motion does not meet those limits, the equations do not apply in the same way.

Rotational Kinematics Using Fixed-Axis Motion

rotational-kinematics-using-fixed-axis-motion

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...

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

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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

Learn how rotational kinematics links angle, angular velocity, angular acceleration, and time for fixed-axis motion with constant angular acceleration.

Rotational Kinematics Using Fixed-Axis Motion | 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

The second kinematic equation expresses the final position of an object in terms of its initial position, the distance traveled with the initial constant ...

Kinematic Equations - III

The first two kinematic equations have time as a variable, but the third kinematic equation is independent of time. This equation expresses final velocity ...

Kinematic Equations: Problem Solving

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Acceleration Vectors

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Solving Problems in Physics

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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

Uniform circular motion is motion in a circle at a constant speed. Although this is the simplest case of rotational motion, it is very useful for many ...

Angular Velocity and Acceleration

We previously discussed angular velocity for uniform circular motion, however not all motion is uniform. Envision an ice skater spinning with their arms ...

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 ...

Equation of Rotational Dynamics

Angular variables are introduced in rotational dynamics. Comparing the definitions of angular variables with the definitions of linear kinematic ...

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Relating Angular And Linear Quantities - I

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

Relating Angular And Linear Quantities - II

In the case of circular motion, the linear tangential speed of a particle at a radius from the axis of rotation is related to the angular velocity by the ...

Moment of Inertia

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

Moment of Inertia and Rotational Kinetic Energy

The rotational kinetic energy of a body is equal to half the square of its angular speed and the moment of inertia.
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

Forces affect every moment of our life. Our bodies are held to the Earth by force, and they are held together by the forces of charged particles. When we ...

Types of Forces

In most situations, forces can be grouped into two categories: contact forces and field forces.&#160; Contact forces occur as a result of direct physical ...

Newton's First Law: Introduction

Motion draws our attention. Motion itself can be beautiful, causing us to marvel at the forces needed to create spectacular sights, such as that of a ...

Newton's First Law: Application

Experience suggests that an object at rest remains at rest if left alone, and that an object in motion tends to slow down and stop unless some effort is ...

Newton's Second Law

Newton&#39;s second law is closely related to his first law of motion. It mathematically gives the cause-and-effect relationship between force and changes ...

Newton's Third Law: Introduction

Whenever one body exerts a force on a second body, the first body experiences a force equal in magnitude and opposite in direction, to the force that it ...

Newton's Third Law: Examples

Newton&#39;s third law states that every action has an equal and opposite reaction. Consider a swimmer pushing off the side of a pool. They push against ...

Drawing Free-body Diagrams: Rules

The first step in describing and analyzing most phenomena in physics involves the careful drawing of a free-body diagram. Free-body diagrams are useful in ...

Free Body Diagrams: Examples

Solving problems that involve forces is easy using free-body diagrams. A free-body diagram is a sketch showing all the external forces that are acting on ...

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

Vector Components in the Cartesian Coordinate System

Vectors are usually described in terms of their components in a coordinate system. Even in everyday life, we naturally invoke the concept of orthogonal ...

Polar and Cylindrical Coordinates

The Cartesian coordinate system is a very convenient tool to use when describing the displacements and velocities of objects and the forces acting on ...

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Spherical Coordinates

Spherical coordinate systems are preferred over Cartesian, polar, or cylindrical coordinates for systems with spherical symmetry. For example, to describe ...

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Vector Algebra: Graphical Method

Vectors can be multiplied by scalars, added to other vectors, or subtracted from other vectors. The vector sum of two (or more) vectors is called the ...

Vector Algebra: Method of Components

It is cumbersome to find the magnitudes of vectors using the parallelogram rule or using the graphical method to perform mathematical operations like ...

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Work Done Over an Inclined Plane

The center-of-mass framework helps to easily describe the work done on rigid bodies. Since the internal forces in a rigid body do no work, they can be ...

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Projectile Motion

An object thrown in the air follows a parabolic path under the influence of Earth&#39;s gravitational force. The motion of such an object is called ...

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Projectile Motion: Equations

Projectile motion is commonly observed in our day-to-day life. For example, a basketball thrown by a player, an arrow shot from a bow, and kids jumping ...

Projectile Motion: Example

The theory of projectile motion is very useful for players of several sports to improve their performance. For example, a javelin thrower needs to throw ...

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

Mass and weight are often used interchangeably in everyday conversation. For example,&#160; medical records often show our weight in kilograms, but never ...

Gravitation

In the years before Newton, a general belief prevailed that different laws governed objects in the sky than objects on Earth. When Kepler wrote down the ...

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Newton's Law of Gravitation

Our everyday observation tells us that all objects close to the Earth naturally tend to fall to the ground. Early philosophers assumed that this downward ...

Free-falling Bodies: Introduction

All objects, neglecting air resistance, fall with the same acceleration towards the Earth&#39;s center due to the force exerted by the Earth&#39;s ...

Free-falling Bodies: Example

An object falling without any air resistance under the influence of gravitational force is said to be in free-fall. For free-falling bodies, the ...

Acceleration due to Gravity on Earth

According to Newton&#39;s law of gravitation, the gravitational force on a body is proportional to its mass. According to Newton&#39;s second law of ...

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

The Principle of Superposition and the Gravitational Field

The principle of superposition applies to gravitational forces of objects that are sufficiently far apart. It states that the net gravitational force on a ...

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Hooke's Law and Simple Harmonic Motion

Source: Ketron Mitchell-Wynne, PhD,&#160;Asantha Cooray, PhD, Department of Physics &#38; Astronomy, School of Physical Sciences, University of ...

Hooke's Law

Hooke&#39;s law states that stress applied to a material is directly proportional to the strain it experiences within the elastic limit. For ductile ...

Frequency of Spring-Mass System

One interesting characteristic of the simple harmonic motion (SHM) of an object attached to a spring is that the angular frequency, and the period and ...

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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

Newton&#39;s laws can be applied to bodies at rest and bodies in motion. Newton&#39;s first law is applied to bodies in equilibrium, whereas the second ...

second-law-motion-under-same-acceleration

Second Law: Motion under Same Acceleration

Newton&#39;s second law of motion applies to bodies moving under the same acceleration. For example, when a baggage tractor pulls luggage carts, each cart ...

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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Conservation of Momentum: Introduction

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Conservation of Momentum: Problem Solving

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Types Of Collisions - I

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Types of Collisions - II

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Elastic Collisions: Introduction

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Elastic Collisions: Case Study

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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 Kinematics Using Fixed-Axis Motion

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