




Check Your Understanding Which has greater angular momentum: a solid sphere of mass
m rotating at a constant angular frequency
${\omega}_{0}$ about the
z axis, or a solid cylinder of same mass and rotation rate about the
z axis?
${I}_{\text{sphere}}=\frac{2}{5}m{r}^{2},\phantom{\rule{0.5em}{0ex}}{I}_{\text{cylinder}}=\frac{1}{2}m{r}^{2}$ ; Taking the ratio of the angular momenta, we have:
$\frac{{L}_{\text{cylinder}}}{{L}_{\text{sphere}}}=\frac{{I}_{\text{cylinder}}{\omega}_{0}}{{I}_{\text{sphere}}{\omega}_{0}}=\frac{\frac{1}{2}m{r}^{2}}{\frac{2}{5}m{r}^{2}}=\frac{5}{4}$ . 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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Summary
 The angular momentum
$\overrightarrow{l}=\overrightarrow{r}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\overrightarrow{p}$ 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
$\overrightarrow{l}={\displaystyle \sum _{i}{\overrightarrow{l}}_{i}}$ 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:
$\frac{d\overrightarrow{L}}{dt}={\displaystyle \sum \overrightarrow{\tau}}$ .
 A rigid rotating body has angular momentum
$L=I\omega $ directed along the axis of rotation. The time derivative of the angular momentum
$\frac{dL}{dt}={\displaystyle \sum \tau}$ gives the net torque on a rigid body and is directed along the axis of rotation.
Conceptual questions
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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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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Problems
A 0.2kg particle is travelling along the line
$y=2.0\phantom{\rule{0.2em}{0ex}}\text{m}$ with a velocity
$5.0\phantom{\rule{0.2em}{0ex}}\text{m}\text{/}\text{s}$ . 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
$\theta $ 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
$rp\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\theta $ , which gives
$r\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\theta $ 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
$\widehat{k}$ , which is in the plane of the ground:
$\overrightarrow{L}=\overrightarrow{r}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\overrightarrow{p}=hmv\widehat{k}=(300.0\phantom{\rule{0.2em}{0ex}}\text{m})(2.0\phantom{\rule{0.2em}{0ex}}\text{kg})(20.0\phantom{\rule{0.2em}{0ex}}\text{m}\text{/}\text{s})\widehat{k}=\mathrm{12,000.0}\phantom{\rule{0.2em}{0ex}}\text{kg}\xb7{\text{m}}^{2}\text{/}\text{s}\widehat{k}$
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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
a particle projected from origin moving on xy plane passes through P & Q having consituents (9,7) , (18,4) respectively.find eq. of trajectry.
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.
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 crosssectional 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
hello are you ready to ask aquestion?
what is binary operations
Tollum
What is the formula to calculat parallel forces that acts in opposite direction?
position, velocity and acceleration of vector
*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
A rail way truck of mass 2400kg is hung onto a stationary trunk on a level track and collides with it at 4.7ms. After collision the two trunk move together with a common speed of 1.2ms. Calculate the mass of the stationary trunk
I need the solving for this question
philip
is the eye the same like the camera
I can't understand
Suraia
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?
why classical mechanics is necessary for graduate students?
classical mechanics?
Victor
principle of superposition?
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?
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
(10/6) ÷0.4=4.167 per sec
Shubhrant
what is the formula for pressure?
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
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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