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Check Your Understanding Which has greater angular momentum: a solid sphere of mass m rotating at a constant angular frequency ω 0 about the z -axis, or a solid cylinder of same mass and rotation rate about the z -axis?

I sphere = 2 5 m r 2 , I cylinder = 1 2 m r 2 ; Taking the ratio of the angular momenta, we have:
L cylinder L sphere = I cylinder ω 0 I sphere ω 0 = 1 2 m r 2 2 5 m r 2 = 5 4 . Thus, the cylinder has 25 % more angular momentum. This is because the cylinder has more mass distributed farther from the axis of rotation.

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Visit the University of Colorado’s Interactive Simulation of Angular Momentum to learn more about angular momentum.

Summary

  • The angular momentum l = r × p of a single particle about a designated origin is the vector product of the position vector in the given coordinate system and the particle’s linear momentum.
  • The angular momentum l = i l i of a system of particles about a designated origin is the vector sum of the individual momenta of the particles that make up the system.
  • The net torque on a system about a given origin is the time derivative of the angular momentum about that origin: d L d t = τ .
  • A rigid rotating body has angular momentum L = I ω directed along the axis of rotation. The time derivative of the angular momentum d L d t = τ gives the net torque on a rigid body and is directed along the axis of rotation.

Conceptual questions

Can you assign an angular momentum to a particle without first defining a reference point?

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For a particle traveling in a straight line, are there any points about which the angular momentum is zero? Assume the line intersects the origin.

All points on the straight line will give zero angular momentum, because a vector crossed into a parallel vector is zero.

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Under what conditions does a rigid body have angular momentum but not linear momentum?

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If a particle is moving with respect to a chosen origin it has linear momentum. What conditions must exist for this particle’s angular momentum to be zero about the chosen origin?

The particle must be moving on a straight line that passes through the chosen origin.

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If you know the velocity of a particle, can you say anything about the particle’s angular momentum?

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Problems

A 0.2-kg particle is travelling along the line y = 2.0 m with a velocity 5.0 m / s . What is the angular momentum of the particle about the origin?

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A bird flies overhead from where you stand at an altitude of 300.0 m and at a speed horizontal to the ground of 20.0 m/s. The bird has a mass of 2.0 kg. The radius vector to the bird makes an angle θ with respect to the ground. The radius vector to the bird and its momentum vector lie in the xy -plane. What is the bird’s angular momentum about the point where you are standing?

The magnitude of the cross product of the radius to the bird and its momentum vector yields r p sin θ , which gives r sin θ as the altitude of the bird h . The direction of the angular momentum is perpendicular to the radius and momentum vectors, which we choose arbitrarily as k ^ , which is in the plane of the ground:
L = r × p = h m v k ^ = ( 300.0 m ) ( 2.0 kg ) ( 20.0 m / s ) k ^ = 12,000.0 kg · m 2 / s k ^

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A Formula One race car with mass 750.0 kg is speeding through a course in Monaco and enters a circular turn at 220.0 km/h in the counterclockwise direction about the origin of the circle. At another part of the course, the car enters a second circular turn at 180 km/h also in the counterclockwise direction. If the radius of curvature of the first turn is 130.0 m and that of the second is 100.0 m, compare the angular momenta of the race car in each turn taken about the origin of the circular turn.

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Questions & Answers

What is angular velocity
Amara Reply
An object undergoes constant acceleration after starting from rest and then travels 5m in the first seconds .determine how far it will go in the next seconds
Sophy Reply
x=x0+v0t+1/2at^2
Grant
what is the deference between precision and accuracy
ShAmy
In measurement of a set, accuracy refers to closeness of the measurements to a specific value, while precision refers to the closeness of the measurements to each other. 
Grant
Thank you Mr..
ShAmy
Are there is a difference between Error and Relative Error...
ShAmy
An aircraft flies 300km due east and 600km due north. determine the magnitude of its displacement
Sophy Reply
670.8km
iyiola
which formula did you use
Sophy
X^2=300^2+600^2
iyiola
what is Fermat principle.
Madhu Reply
find the angle of projection at which the horizontal range is twice the maximum height of a projectile
Akaji Reply
impulse by height fomula
Tigst Reply
Hello
Madue
what is impulse?
Tigst Reply
impulse is the integral of a force (F),over the interval for which it act
Agbeyangi
What is significance of vector?
jagan Reply
magnitude and direction
Chris
what is impulse?
Enimini Reply
the product of force and time
Robert
find the flow rate of a fluid of viscosity 0.0015N-s/m³ flowing through a pipe of radius 50cm and length 100m at a pressure differential of 200000N/m²
Okoli Reply
friend ,what is the equation your question
lasitha
What is victor
Moses Reply
How are you all?
Giorgi Reply
what is the formula for momentum?
Chisom Reply
p=mv
Grant
what is the formula for angular displacement ?
lasitha
angular displacement = l/r
anand
friend ,what is the " l "?
lasitha
I have another problem,what is the difference between pure chemistry and applied chemistry ?
lasitha
l stands for linear displacement and l>>r
anand
r stands for radius or position vector of particle
anand
pure chemistry is related to classical and basic concepts of chemistry while applied Chemistry deals with it's application oriented concepts and procedures.
anand
what is the deference between precision and accuracy?
ShAmy
I think in general both are same , more accuracy means more precise
anand
and lessor error
anand
friends,how to find correct applied chemistry notes?
lasitha
what are the main concepts of applied chemistry ?
lasitha
anand,can you give an example for angular displacement ?
lasitha
when any particle rotates or revolves about something , an axis or point ,then angle traversed is angular displacement
anand
like a revolving fan
anand
yeah ,I understood now, thanks!
lasitha
o need this application in French language
Diacre Reply
Why is it that the definition is not there
Bello
what is physics
mesfin Reply
the study of matter in relation to energy
Robert
Ampher law
Zia
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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