Name: center of mass and particle momentum
Uploaded: 2024-03-07T13:22:20.000Z
Duration: 1 min 21 s
Channel: JoVE

K12 Standards

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South Carolina College and Career Ready Science Standards

High School

Physics

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Motion and Stability: Forces and Interactions

P-PS2-2

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

Principle of Linear Impulse and Momentum for a System of Particles

Consider a system of particles moving relative to an inertial frame of reference.
The equation of motion for such a system can be written as the sum of external forces acting on each particle.
Since the internal forces between particles occur in equal and opposite collinear pair, they are excluded from the equation.
Integrating the equation of motion and substituting the limits yields the equation for the principle of linear impulse and momentum for the system of particles.
This principle states that the initial linear momentum of the system, combined with the impulses of all external forces from the initial to the final time, equals the final linear momentum of the system.
Now, consider the equation of the system&#39;s center of mass. Differentiating it, the total linear momentum of the particles can be related to the linear momentum of the center of mass.
This relationship is then substituted into the equation for linear impulse and momentum.
The modified equation implies the applicability of the principle to the system of particles that compose a rigid body.

principle of linear impulse and momentum for a system of particles

In the context of a system of particles moving relative to an inertial frame of reference, the equation of motion is a crucial tool for understanding the dynamics of the system. This equation, which accounts for external forces acting on each particle, plays a fundamental role in describing the system&#39;s behavior.
Notably, internal forces between particles, occurring in equal and opposite collinear pairs, cancel out and are not part of the equation of motion. This exclusion simplifies the analysis, focusing on the impact of external forces on the system.
The principle of linear impulse and momentum for the system of particles emerges upon integrating the equation of motion and substituting the limits. According to this principle, the combined initial linear momenta of the particles, along with the impulses of all external forces from the initial to the final time, equals the final linear momenta of the system.
To extend the applicability of this principle, we delve into the equation governing the system&#39;s center of mass. By differentiating it, a relationship between the total linear momentum of the particles and the linear momentum of the center of mass is established. This crucial relationship is then incorporated back into the linear impulse and momentum equation, resulting in a modified equation that aptly applies to the broader context of a system of particles composing a rigid body. This nuanced approach enhances our understanding of the dynamic interactions within such systems.

Consider a system of particles moving relative to an inertial frame of reference.
The equation of motion for such a system can be written as the sum of ...

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

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 the particles plus the impulses of all external forces from the initial time to the final time equal the final linear momenta of the system. This gives a compact way to track motion when forces act over a time interval.

The center of mass helps extend the same idea to a wider case. By differentiating the center-of-mass equation, a link is found between the total linear momentum of the particles and the linear momentum of the center of mass. That relationship can then be placed back into the linear impulse and momentum equation to form a modified version that applies to a system of particles making up a rigid body.

Center of Mass and Particle Momentum

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...

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...

relation between moment of a force and angular momentum

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

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...

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

Learn how external forces, impulse, and momentum work for a system of particles, and how the center of mass leads to a rigid-body form of the equation.

Center of Mass and Particle Momentum | 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

P-PS2-3

Newton's Second Law

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Force and Acceleration

Source: Nicholas Timmons,&#160;Asantha Cooray, PhD, Department of Physics &#38; Astronomy, School of Physical Sciences, University of California, Irvine, ...

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 ...

Newton's Laws of Motion

Source: Andrew Duffy, PhD, Department of Physics, Boston University, Boston, MA
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Newton's Law of Motion

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Analyze data to support the claim that Newton's second law of motion describes the mathematical relationship among the net force on a macroscopic object, its mass, and its acceleration.

P-PS2-1

Force and Momentum

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Internal and External Forces

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Force

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Conservation of Momentum: Problem Solving

Solving problems using the conservation of momentum requires four basic steps:

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Conservation of Linear Momentum for a System of Particles

In a game of billiards, when the cue ball strikes a stationary ball, it transfers some of its momentum to the stationary ball. The cue ball slows down, ...

Use mathematical representations to support the claim that the total momentum of a system of objects is conserved when there is no net force on the system.

Principle of Linear Impulse and Momentum for a Single Particle

Linear momentum is a fundamental concept in physics that describes the motion of an object. It is a vector quantity, having a magnitude equal to the ...

Principle of Angular Impulse and Momentum: Problem Solving

Consider a ball of mass m, attached to a massless rod of known length, subjected to a time-dependent torque. If the initial velocity of the mass is known, ...

Linear Momentum

The term momentum is used in various ways in everyday language, most of which are consistent with the precise scientific definition. Generally, momentum ...

Principle of Angular Impulse and Momentum

The angular impulse and momentum principle provides insights into how forces applied at a distance from an object&#39;s rotational axis influence its ...

Principle of Linear Impulse and Momentum for a Single Particle: Problem Solving

Consider a massless pulley system consisting of a wooden box and a cylinder of known masses. Determine the speed of the attached objects at a specific ...

Angular Momentum

Angular momentum describes the rotational motion of an object.
It is defined as the moment of the object&#39;s linear momentum about a specific point O.
...

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 ...

Apply scientific and engineering ideas to design, evaluate, and refine a device that minimizes the effect of a force on a macroscopic object during a collision.

Gravitation

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Gravitation Between Spherically Symmetric Masses

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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, ...

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comparison-between-electrical-and-gravitational-forces

Comparison Between Electrical And Gravitational Forces

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Newton's Law of Universal Gravitation

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P-PS2-4

Types of Forces

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Magnetic Fields

Source: Yong P. Chen, PhD, Department of Physics &#38; Astronomy, College of Science, Purdue University, West Lafayette, IN
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Electric Charge in a Magnetic Field

Source: Andrew Duffy, PhD, Department of Physics, Boston University, Boston, MA
This experiment duplicates J.J. Thomson's famous experiment at the end of ...

Electric Fields

Source: Yong P. Chen, PhD, Department of Physics &#38; Astronomy, College of Science, Purdue University, West Lafayette, IN
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Electromagnetic Fields

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Magnetic Field Due to Two Straight Wires

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Energy Carried By Electromagnetic Waves

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Energy Stored in Capacitors

A parallel plate capacitor connected to a battery develops a potential difference across its plates.
By integrating the equation relating voltage and ...

Power and Energy

Energy is the total work required to drive the current through the circuit.
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Voltage

The movement of electrons in a conductor requires some form of energy or work, usually provided by an external force, like a battery. This force is called ...

Electromotive Force

Electromotive force (emf) is the force that causes current to flow from a higher to a lower&#160; potential. The term &#34;electromotive ...

Electrical Energy

Using electric appliances for a longer period of time consumes more electrical energy and results in a higher electric bill. The energy produced by the ...

Electrical Current

Electrical current is defined as the rate at which charge flows. When there is a large current present, such as that used to run a refrigerator, a large ...

Electric Potential Energy

When an electric field accelerates a free positive charge q, it is given kinetic energy. The process is analogous to an object accelerated by a ...

Electrical Power

Electric power is the product of current and voltage, represented in units of joules per second, or watts. For example, cars often have one or more ...

Plan and conduct an investigation to provide evidence that an electric current can produce a magnetic field and that a changing magnetic field can produce an electric current.

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Sources and Properties of Electric Charge

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Electric Charges

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Consider two point charges, each exerting Coulomb force on the other. It is possible to describe the Coulomb interaction via an intermediate step by ...

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Electric Field of Two Equal and Opposite Charges

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Communicate scientific and technical information about why the molecular-level structure is important in the functioning of designed materials.

P-PS2-6.

Center of Mass and Particle Momentum

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