Name: angular acceleration in rotational motion
Uploaded: 2025-05-22T09:05:40.000Z
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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

Angular Velocity and Acceleration

Consider a ball tied to a string rotating in a circular trajectory. The rate of change of angular displacement is called its angular velocity. Like linear velocity, angular velocity is also a vector quantity, and a rotation in a clockwise direction is considered as the negative direction. Consider a ball rotating at high speed, the rate of change of angular displacement will be high, and hence the angular velocity will be high.
The angular velocity value at any time during the motion is called its instantaneous angular velocity, and it is expressed as a derivative of &#952; with respect to time.
The rate at which the angular velocity of an object changes is called its angular acceleration, denoted by the letter &#945; with units of radians per second square.
Angular acceleration is a vector quantity and is considered positive when the angular velocity increases and vice-versa. Since two points on the string have the same angular velocity, their angular acceleration is also the same.

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 outstretched; when they pull their arms inward, their angular velocity increases. Additionally, think about a computer&#39;s hard disk slowing to a halt as the angular velocity decreases. The faster the change in angular velocity, the greater the angular acceleration. The instantaneous angular acceleration is defined as the derivative of angular velocity with respect to time. The units of angular acceleration are (rad/s)/s, or radians per second squared.
We can relate the tangential acceleration of a point on a rotating body at a distance from the axis of rotation in the same way that we relate the tangential speed to the angular velocity. Thus, tangential acceleration is the radius times the angular acceleration.
The following points represent a problem-solving strategy that can be applied to rotational kinematics:
<ol>
	<li>Examine the situation to determine that rotational kinematics (rotational motion) is involved.</li>
	<li>Identify exactly what needs to be determined in the problem (identify the unknowns). A sketch of the situation is useful.</li>
	<li>Make a complete list of what is given or can be inferred from the problem as stated (identify the knowns).</li>
	<li>Solve the appropriate equation or equations for the quantity to be determined (the unknown). It can be useful to think in terms of a translational analog. Substitute the known values along with their units into the appropriate equation and obtain numerical solutions complete with units. Be sure to use units of radians for angles.</li>
	<li>Finally, check the answer to see if it is reasonable.</li>
</ol>
This text is adapted from <a href="https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables">Openstax, University Physics Volume 1, Section 10.1: Rotational Variables.</a>

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

Education

JoVE Core

Unit 1

Rotation and Rigid Bodies

KEY TERMS AND CONCEPTS

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. Its units are radians per second squared, written as (rad/s)/s.

Angular acceleration also connects to tangential acceleration for a point on a rotating body. Tangential acceleration is the radius times the angular acceleration, just as tangential speed is related to angular velocity by radius. When solving rotational kinematics problems, it helps to first decide that rotational motion is involved, identify the unknowns, and sketch the situation.

Next, list what is known or can be inferred. Then solve the needed equation or equations for the unknown quantity. It can help to compare the problem with a translational analog. Substitute the values with units, use radians for angles, and check that the final answer is reasonable.

Angular Acceleration in Rotational Motion

angular-acceleration-in-rotational-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 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,...

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 angular acceleration, its units, and how it relates to tangential acceleration. Learn a clear problem-solving strategy for rotational kinematics.

Angular Acceleration in Rotational Motion | JoVE K12

physics

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PHYS.PS2.4

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PHYS.PS2.6

Angular Acceleration in Rotational Motion

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