Name: torque and angular momentum in rotation
Uploaded: 2024-03-07T13:22:21.000Z
Duration: 1 min 21 s
Channel: JoVE

K12 Standards

Florida Department of Education

High School

Physical Science

SC.912.P.12

SC.912.P.12.6

relation-between-moment-of-a-force-and-angular-momentum

Relation Between Moment of a Force and Angular Momentum

In a spinning top, applying a force at a distance from the center creates torque, which changes the angular momentum of the top, causing it to spin.
The moment of a force is the rotational equivalent of linear force. It measures how a force on an object induces rotation.
Angular momentum is the rotational analog of linear momentum and signifies the tendency of an object to continue rotating.
The time derivative of angular momentum gives an expression, in which the first term equals zero. The second term can be expressed in terms of the net force acting on the particle.
Comparing the expression with the moment of force equation establishes a relation between the angular momentum and the moment of force.
This relationship mirrors Newton&#39;s second law but for rotational motion.
This equation applies equally to the systems of particles or rigid bodies.
In a system of particles, each particle contributes to the overall angular momentum of the system.

relation between moment of a force and angular momentum

In the realm of spinning tops, the application of force at a distance from the center produces torque, a pivotal factor that alters the angular momentum of the top, thereby inducing its rotation. The concept of moment, akin to linear force in rotation, quantifies how a force acting upon an object initiates rotational motion. Angular momentum serves as the rotational counterpart to linear momentum, representing an object&#39;s inherent tendency to persist in its rotational state.
The temporal change in angular momentum, when expressed as a derivative, yields an equation where the initial term is null, and the subsequent term correlates with the net force acting on the particle. A comparison of this expression with the moment of force equation establishes a crucial link between angular momentum and the moment of force, reflecting a rotational analog of Newton&#39;s second law.
This fundamental equation is universally applicable to both systems of particles and rigid bodies. Within a system of particles, the cumulative angular momentum arises from the individual contributions of each particle. In essence, this conceptual framework extends the principles of Newtonian dynamics to rotational motion, encapsulating the interplay between force, torque, and angular momentum in the dynamic world of rotating objects.

In a spinning top, applying a force at a distance from the center creates torque, which changes the angular momentum of the top, causing it to spin.
The ...

Education

JoVE Core

Mechanical Engineering

Unit 1

Kinetics of a Particle: Impulse and Momentum

kinetics-of-a-particle-impulse-and-momentum

KEY TERMS AND CONCEPTS

torque and angular momentum in rotation

Torque and angular momentum explain how force changes rotation. When a force acts at a distance from the center, it creates torque. A spinning top is a simple example of this idea.

Moment is the rotational form of force. It shows how a force can start rotational motion. Angular momentum is the rotational form of linear momentum. It describes an object’s tendency to keep rotating.

The rate of change of angular momentum can be written as a derivative. In that expression, the first term becomes zero. The next term matches the net force acting on the particle. Comparing this result with the moment of force equation shows the link between torque and angular momentum.

This link is a rotational version of Newton’s second law. It applies to a system of particles and to rigid bodies. In a system of particles, total angular momentum is the sum of the angular momentum of each particle. Together, these ideas describe how force, torque, and angular momentum work in rotating objects.

Torque and Angular Momentum in Rotation

principle of linear impulse and momentum for a single particle

principle-of-linear-impulse-and-momentum-for-a-single-particle

Momentum and Impulse Diagrams for a Particle

Linear momentum describes how a particle moves in physics. It is a vector quantity, so it has both size and direction. Its magnitude equals mass times velocity, and its direction matches the particle’s velocity. Linear impulse, also called momentum impulse, describes the change in linear momentum. It is also a vector quantity. Impulse equals force multiplied by the time interval over which the force acts. For a particle of mass m in an inertial frame, the equation of motion can be integrated...

principle of linear impulse and momentum for a single particle: problem solving

principle-linear-impulse-momentum-for-single-particle-problem

Solving a Two-Mass Pulley Momentum Problem

A massless pulley system with a wooden box and a cylinder shows how linear impulse and momentum can be used to find motion after release. The two objects have known masses, m1 and m2, and the system starts from rest before it is released. The rope length stays constant as the box and cylinder move. That means the rope can be written as a set of smaller segments attached to each object. Taking the time derivative of the full rope length gives the velocity relation between the wooden box and the...

principle of linear impulse and momentum for a system of particles

principle-of-linear-impulse-and-momentum-for-a-system-of-particles

Center of Mass and Particle Momentum

A system of particles follows an equation of motion that describes how external forces affect its motion in an inertial frame of reference. The particles also exert internal forces on one another, but these forces act in equal and opposite collinear pairs. Because they cancel, they do not appear in the equation of motion. The principle of linear impulse and momentum comes from integrating that equation of motion over time. After the limits are applied, the combined initial linear momenta of...

conservation of linear momentum for a system of particles

conservation-of-linear-momentum-for-a-system-of-particles

Momentum Transfer in Billiard Collisions

Momentum transfer in billiard collisions shows how linear momentum changes when one ball strikes another. A moving cue ball hits a stationary ball and gives up part of its linear momentum. The cue ball slows down, and the other ball starts moving. The collision force acts for an extremely short time, so its impulse is negligible. In impulse-momentum analysis, forces are grouped as impulsive or non-impulsive. Impulsive forces act briefly but can cause a large momentum change. For a system of...

impact

Impact, Impulse, and Restitution

Impact describes what happens when two bodies collide and impulsive forces act between them. In impact mechanics, the colliding particles move along the line of impact. This line passes through their centers and is perpendicular to the contact plane. When particles with different initial velocities collide, they apply equal and opposite impulses and cause deformation. At maximum deformation, the particles move together with the same velocity. After that, restitution begins. Restitution is the...

types of impact

Central and Oblique Collision Dynamics

Impact events are described by how two objects collide and how they move after the collision. The two main forms are central impact and oblique impact. A central impact happens when objects strike head-on, with opposite velocities along the line of impact. An oblique impact happens when objects collide at an angle. In this case, both direction and velocity can change after the collision. This makes the motion after impact more complex than in a head-on collision. The coefficient of...

impact: problem solving

Mars Projectile Rebound Height Calculation

Mars projectile rebound height calculation uses kinematics and the coefficient of restitution. A rover launches a projectile, and the projectile hits the Martian surface before bouncing upward again. The goal is to find the maximum height it reaches after the collision. The launch point is taken as the origin. Upward velocity is treated as positive, and the horizontal velocity stays constant. Using the known initial velocity and Mars gravity, the vertical velocity at impact is calculated with...

angular momentum

Angular Momentum and Direction

Angular momentum describes rotational motion and the direction of that motion. It is defined as the moment of a particle’s linear momentum about a chosen point O. This makes it useful for describing how motion curves around a point. For a particle moving along a curved path in the x-y plane, the scalar form gives the magnitude of angular momentum. It uses the moment arm, d, which is the perpendicular distance from point O to the line of action of the linear momentum. Even when written in...

Torque and angular momentum explain how force changes rotation. When a force acts at a distance from the center, it creates torque. A spinning top is a simple example of this idea. Moment is the rotational form of force. It shows how a force can start rotational motion. Angular momentum is the rotational form of linear momentum. It describes an object’s tendency to keep rotating. The rate of change of angular momentum can be written as a derivative. In that expression, the first term becomes...

principle of angular impulse and momentum

principle-of-angular-impulse-and-momentum

Angular Impulse, Momentum, and Rotation

Angular impulse and momentum describe how forces change an object's rotation over time. The idea connects the moment of force, or torque, with angular momentum. When the equation is integrated from an initial time to a final time, it gives the angular impulse and momentum principle. The principle defines initial and final angular momentum as the moments of a particle's linear momentum at those specific times. Angular impulse is found by integrating the moments of all forces acting on the...

principle of angular impulse and momentum: problem solving

principle-of-angular-impulse-and-momentum-problem-solving

Angular Impulse and Final Velocity

Angular impulse and momentum help determine the final velocity of a ball attached to a massless rod. The ball has mass m and moves under a time-dependent torque. If the initial velocity is known, the final velocity after a time interval can be found with this principle. The first step is to draw a free-body diagram. A free-body diagram shows all the forces acting on the system. It helps make the motion and rotation easier to analyze. Next, the principle of angular impulse and momentum is...

steady flow of a fluid stream

Fluid Force and Momentum in a Pipe

Fluid force and momentum in a pipe are linked through a control volume with solid boundaries. As fluid moves through the pipe, the walls can change its direction by exerting an impulse through the resulting force. This setup helps show how flow behaves inside a fixed region. For steady flow, the mass entering the control volume at time t with velocity v1 equals the mass leaving after a small time dt with velocity v2. Over that same interval, the momentum inside the control volume stays...

key terms and concepts

Explore how torque changes angular momentum, how moment relates to rotation, and why this link acts like Newton’s second law for spinning objects.

Torque and Angular Momentum in Rotation | JoVE K12

JoVE Core Mechanical Engineering offers concise, easy-to-understand animated videos covering key topics such as equilibrium, forces, truss analysis, and structural mechanics. Explore their real-world applications, and witness these concepts in action through scientist-led experiments.

mechanical engineering

south-carolina-college-and-career-ready-science-standards

South Carolina College and Career Ready Science Standards

Physics

motion-and-stability-forces-and-interactions

Motion and Stability: Forces and Interactions

P-PS2-3

Introduction to Scalars

Many familiar physical quantities can be specified completely by giving a single number and the appropriate unit. For example, &#34;a class period lasts ...

Introduction to Vectors

To define some physical quantities, there is a need to specify both magnitude as well as direction. For example, when the U.S. Coast Guard dispatches a ...

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 ...

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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Scalar Product (Dot Product)

The scalar multiplication of two vectors is known as the scalar or dot product. As the name indicates, the scalar product of two vectors results in a ...

Vector Product (Cross Product)

Vector multiplication of two vectors yields a vector product, with the magnitude equal to the product of the individual vectors multiplied by the sine of ...

Scalar and Vector Triple Products

Two vectors can be multiplied using a scalar product or a vector product. The resultant of a scalar product is scalar, while with vector products, the ...

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Gradient and Del Operator

In mathematics and physics, the gradient and del operator are fundamental concepts used to describe the behavior of functions and fields in space. The ...

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Divergence and Curl

The divergence of a vector field at a point is the net outward flow of the flux out of a small volume through a closed surface enclosing the volume, as ...

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Second Derivatives and Laplace Operator

The first order operators using the del operator include the gradient, divergence and curl. Certain combinations of first order operators on a scalar or ...

/files/media/science-education/science-education-thumbs/14692.jpg

Line, Surface, and Volume Integrals

A line integral for a vector field is defined as the integral of the dot product of a vector function with an infinitesimal displacement vector along a ...

/files/media/science-education/science-education-thumbs/14693.jpg

Distinguish between scalar and vector quantities and assess which should be used to describe an event.

SC.912.P.12.1

Position and Displacement

The position of an object defines its location relative to a convenient frame of reference at any particular time. A frame of reference is an arbitrary ...

Average Velocity

To calculate the other physical quantities in kinematics, we must introduce the time variable. The time variable allows us not only to state the position ...

Instantaneous Velocity - I

The average velocity during a time interval cannot tell us how fast or in what direction a particle is moving at any given time during the interval. To ...

Instantaneous Velocity - II

Instantaneous velocity is the quantity that measures how fast an object is moving along its path. In other words, the instantaneous velocity of an object ...

Average Acceleration

The importance of understanding acceleration spans our day-to-day experiences, as well as the vast reaches of outer space and the tiny world of subatomic ...

Instantaneous Acceleration

Acceleration is in the direction of the change in velocity, but it is not always in the direction of motion. When an object slows down, its acceleration ...

Kinematic Equations - I

When an object moves with constant acceleration, the velocity of the object changes at a constant rate throughout the motion. The kinematic equations of ...

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

When analyzing one-dimensional motion with constant acceleration, the problem-solving strategy involves identifying the known quantities and choosing the ...

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 ...

velocity-and-position-by-graphical-method

Velocity and Position by Graphical Method

Velocity and position can be calculated from the known function of acceleration as a function of time. The total area under the acceleration-time graph ...

Non-uniform Circular Motion

In&#160;uniform circular motion, the particle executing circular motion has a constant speed, and the circle is at a fixed radius. However, not all ...

Velocity and Position by Integral Method

If acceleration as a function of time is known, then velocity and position functions can be derived using integral calculus. For constant acceleration, ...

Relative Velocity in One Dimension

The understanding of the concept of reference frames is essential to discuss relative motion in one or more dimensions. When we say that an object has a ...

Position and Displacement Vectors

To describe the motion of an object, one should first be able to describe its position (where it is at any particular time). More precisely, the position ...

Relative Velocity in Two Dimensions

Relative velocity is the velocity of an object as observed from a particular reference frame, or the velocity of one reference frame with respect to ...

average-and-instantaneous-velocity-vectors

Average and Instantaneous Velocity Vectors

To calculate other physical quantities in kinematics, the time variable must be introduced. The time variable not only allows us to state where an object ...

Acceleration Vectors

In everyday conversation, accelerating means speeding up. Acceleration is a vector in the same direction as the change in velocity, &#916;v, therefore the ...

Direction of Acceleration Vectors

Acceleration occurs when velocity changes in magnitude (an increase or decrease in speed), direction, or both. Although acceleration is in the direction ...

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 ...

Uniform Circular Motion

Uniform circular motion is a specific type of motion in which an object travels in a circle with a constant speed. For example, any point on a propeller ...

Analyze the motion of an object in terms of its position, velocity, and acceleration (with respect to a frame of reference) as functions of time.

SC.912.P.12.2

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 ...

Internal and External Forces

Newton&#39;s first law states that a net external force causes a change in motion. External forces act on an object or system, originating outside of the ...

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 ...

Inertial Frames of Reference

Newton&#8217;s first law is usually considered to be a statement about reference frames. It provides a method for identifying a special type of reference ...

Non-inertial Frames of Reference

A reference frame accelerating or decelerating relative to an inertial frame is a non-inertial frame. To help understand this, consider what taking off in ...

first-law-particles-in-one-dimensional-equilibrium

First Law: Particles in One-dimensional Equilibrium

Newton&#39;s first law of motion states that a body at rest remains at rest, or if in motion, remains in motion at constant velocity, unless acted on by a ...

first-law-particles-in-two-dimensional-equilibrium

First Law: Particles in Two-dimensional Equilibrium

Recall that a particle in equilibrium is one for which the external forces are balanced. Static equilibrium involves objects at rest, and dynamic ...

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 ...

Interpret and apply Newton's three laws of motion.

SC.912.P.12.3

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 ...

gravitation-between-spherically-symmetric-masses

Gravitation Between Spherically Symmetric Masses

The gravitational potential energy between two spherically symmetric bodies can be calculated from the masses and the distance between the bodies, ...

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Gravity between Spherical Bodies

Newton&#39;s law of gravitation describes the gravitational force between any two point masses. However, for extended spherical objects like the Earth, ...

Describe how the gravitational force between two objects depends on their masses and the distance between them.

SC.912.P.12.4

A. Motion can be measured and described qualitatively and quantitatively. Net forces create a change in motion. When objects travel at speeds comparable to the speed of light, Einstein's special theory of relativity applies.

B. Momentum is conserved under well-defined conditions. A change in momentum occurs when a net force is applied to an object over a time interval.

C. The Law of Universal Gravitation states that gravitational forces act on all objects irrespective of their size and position.

D. Gases consist of great numbers of molecules moving in all directions. The behavior of gases can be modeled by the kinetic molecular theory.

E. Chemical reaction rates change with conditions under which they occur. Chemical equilibrium is a dynamic state in which forward and reverse processes occur at the same rates.

Torque and Angular Momentum in Rotation

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