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Significance

Since the meteor is accelerating downward toward Earth, its radius and velocity vector are changing. Therefore, since l = r × p , the angular momentum is changing as a function of time. The torque on the meteor about the origin, however, is constant, because the lever arm r and the force on the meteor are constants. This example is important in that it illustrates that the angular momentum depends on the choice of origin about which it is calculated. The methods used in this example are also important in developing angular momentum for a system of particles and for a rigid body.

Check Your Understanding A proton spiraling around a magnetic field executes circular motion in the plane of the paper, as shown below. The circular path has a radius of 0.4 m and the proton has velocity 4.0 × 10 6 m / s . What is the angular momentum of the proton about the origin?

A proton moves in a counterclockwise circle. The circle has radius r. The proton is shown when it is to the right of the center of the circle, and its velocity is v sub perpendicular in the upward, positive y, direction.

From the figure, we see that the cross product of the radius vector with the momentum vector gives a vector directed out of the page. Inserting the radius and momentum into the expression for the angular momentum, we have
l = r × p = ( 0.4 m i ^ ) × ( 1.67 × 10 −27 kg ( 4.0 × 10 6 m / s ) j ^ ) = 2.7 × 10 −21 kg · m 2 / s k ^

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Angular momentum of a system of particles

The angular momentum of a system of particles is important in many scientific disciplines, one being astronomy. Consider a spiral galaxy, a rotating island of stars like our own Milky Way. The individual stars can be treated as point particles, each of which has its own angular momentum. The vector sum of the individual angular momenta give the total angular momentum of the galaxy. In this section, we develop the tools with which we can calculate the total angular momentum of a system of particles.

In the preceding section, we introduced the angular momentum of a single particle about a designated origin. The expression for this angular momentum is l = r × p , where the vector r is from the origin to the particle, and p is the particle’s linear momentum. If we have a system of N particles, each with position vector from the origin given by r i and each having momentum p i , then the total angular momentum of the system of particles about the origin is the vector sum of the individual angular momenta about the origin. That is,

L = l 1 + l 2 + + l N .

Similarly, if particle i is subject to a net torque τ i about the origin, then we can find the net torque about the origin due to the system of particles by differentiating [link] :

d L d t = i d l i d t = i τ i .

The sum of the individual torques produces a net external torque on the system, which we designate τ . Thus,

d L d t = τ .

[link] states that the rate of change of the total angular momentum of a system is equal to the net external torque acting on the system when both quantities are measured with respect to a given origin. [link] can be applied to any system that has net angular momentum, including rigid bodies, as discussed in the next section.

Angular momentum of three particles

Referring to [link] (a), determine the total angular momentum due to the three particles about the origin. (b) What is the rate of change of the angular momentum?

Three particles in the x y plane with different position and momentum vectors are shown. The x and y axes show position in meters and have a range of -4.0 to 4.0 meters. Particle 1 is at x=-2.0 meters and y=1.0 meters, m sub 1 equals 2.0 kilograms, v sub 1 is 4.0 j hat meters per second, upward, and F sub 1 is -6.0 i hat Newtons to the left. Particle 2 is at x=4.0 meters and y=1.0 meters, m sub 2 equals 4.0 kilograms, v sub 2 is 5.0 i hat meters per second, to the right, and F sub 2 is 10.0 j hat Newtons up. Particle 3 is at x=2.0 meters and y=-2.0 meters, m sub 3 equals 1.0 kilograms, v sub 3 is 3.0 i hat meters per second, to the right, and F sub 3 is -8.0 j hat Newtons down.
Three particles in the xy- plane with different position and momentum vectors.

Strategy

Write down the position and momentum vectors for the three particles. Calculate the individual angular momenta and add them as vectors to find the total angular momentum. Then do the same for the torques.

Solution

  1. Particle 1: r 1 = −2.0 m i ^ + 1.0 m j ^ , p 1 = 2.0 kg ( 4.0 m / s j ^ ) = 8.0 kg · m / s j ^ ,
    l 1 = r 1 × p 1 = −16.0 kg · m 2 / s k ^ .

    Particle 2: r 2 = 4.0 m i ^ + 1.0 m j ^ , p 2 = 4.0 kg ( 5.0 m / s i ^ ) = 20.0 kg · m / s i ^ ,
    l 2 = r 2 × p 2 = −20.0 kg · m 2 / s k ^ .

    Particle 3: r 3 = 2.0 m i ^ 2.0 m j ^ , p 3 = 1.0 kg ( 3.0 m / s i ^ ) = 3.0 kg · m / s i ^ ,
    l 3 = r 3 × p 3 = 6.0 kg · m 2 / s k ^ .

    We add the individual angular momenta to find the total about the origin:
    l T = l 1 + l 2 + l 3 = −30 kg · m 2 / s k ^ .
  2. The individual forces and lever arms are
    r 1 = 1.0 m j ^ , F 1 = −6.0 N i ^ , τ 1 = 6.0 N · m k ^ r 2 = 4.0 m i ^ , F 2 = 10.0 N j ^ , τ 2 = 40.0 N · m k ^ r 3 = 2.0 m i ^ , F 3 = −8.0 N j ^ , τ 3 = −16.0 N · m k ^ .

    Therefore:
    i τ i = τ 1 + τ 2 + τ 3 = 30 N · m k ^ .

Significance

This example illustrates the superposition principle for angular momentum and torque of a system of particles. Care must be taken when evaluating the radius vectors r i of the particles to calculate the angular momenta, and the lever arms, r i to calculate the torques, as they are completely different quantities.

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Questions & Answers

a particle projected from origin moving on x-y plane passes through P & Q having consituents (9,7) , (18,4) respectively.find eq. of trajectry.
rahul Reply
definition of inertia
philip Reply
the reluctance of a body to start moving when it is at rest and to stop moving when it is in motion
charles
An inherent property by virtue of which the body remains in its pure state or initial state
Kushal
why current is not a vector quantity , whereas it have magnitude as well as direction.
Aniket Reply
why
daniel
the flow of current is not current
fitzgerald
bcoz it doesn't satisfy the algabric laws of vectors
Shiekh
The Electric current can be defined as the dot product of the current density and the differential cross-sectional area vector : ... So the electric current is a scalar quantity . Scalars are related to tensors by the fact that a scalar is a tensor of order or rank zero .
Kushal
what is binomial theorem
Tollum Reply
hello are you ready to ask aquestion?
Saadaq Reply
what is binary operations
Tollum
What is the formula to calculat parallel forces that acts in opposite direction?
Martan Reply
position, velocity and acceleration of vector
Manuel Reply
hi
peter
hi
daniel
hi
Vedisha
*a plane flies with a velocity of 1000km/hr in a direction North60degree east.find it effective velocity in the easterly and northerly direction.*
imam
hello
Lydia
hello Lydia.
Sackson
What is momentum
isijola
hello
Saadaq
A rail way truck of mass 2400kg is hung onto a stationary trunk on a level track and collides with it at 4.7m|s. After collision the two trunk move together with a common speed of 1.2m|s. Calculate the mass of the stationary trunk
Ekuri Reply
I need the solving for this question
philip
is the eye the same like the camera
EDWIN Reply
I can't understand
Suraia
same here please
Josh
I think the question is that ,,, the working principal of eye and camera same or not?
Sardar
yes i think is same as the camera
muhammad
what are the dimensions of surface tension
samsfavor
why is the "_" sign used for a wave to the right instead of to the left?
MUNGWA Reply
why classical mechanics is necessary for graduate students?
khyam Reply
classical mechanics?
Victor
principle of superposition?
Naveen Reply
principle of superposition allows us to find the electric field on a charge by finding the x and y components
Kidus
Two Masses,m and 2m,approach each along a path at right angles to each other .After collision,they stick together and move off at 2m/s at angle 37° to the original direction of the mass m. What where the initial speeds of the two particles
MB
2m & m initial velocity 1.8m/s & 4.8m/s respectively,apply conservation of linear momentum in two perpendicular directions.
Shubhrant
A body on circular orbit makes an angular displacement given by teta(t)=2(t)+5(t)+5.if time t is in seconds calculate the angular velocity at t=2s
MB
2+5+0=7sec differentiate above equation w.r.t time, as angular velocity is rate of change of angular displacement.
Shubhrant
Ok i got a question I'm not asking how gravity works. I would like to know why gravity works. like why is gravity the way it is. What is the true nature of gravity?
Daniel Reply
gravity pulls towards a mass...like every object is pulled towards earth
Ashok
An automobile traveling with an initial velocity of 25m/s is accelerated to 35m/s in 6s,the wheel of the automobile is 80cm in diameter. find * The angular acceleration
Goodness Reply
(10/6) ÷0.4=4.167 per sec
Shubhrant
what is the formula for pressure?
Goodness Reply
force/area
Kidus
force is newtom
Kidus
and area is meter squared
Kidus
so in SI units pressure is N/m^2
Kidus
In customary United States units pressure is lb/in^2. pound per square inch
Kidus
Practice Key Terms 1

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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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