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A particle of mass 5.0 kg has position vector r = ( 2.0 i ^ 3.0 j ^ ) m at a particular instant of time when its velocity is v = ( 3.0 i ^ ) m / s with respect to the origin. (a) What is the angular momentum of the particle? (b) If a force F = 5.0 j ^ N acts on the particle at this instant, what is the torque about the origin?

a. l = 45.0 kg · m 2 / s k ^ ;
b. τ = 10.0 N · m k ^

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Use the right-hand rule to determine the directions of the angular momenta about the origin of the particles as shown below. The z- axis is out of the page.

Fout particles in the x y plane with different position and velocity 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 has mass m sub 1, is at x=0 meters and y=2.0 meters, and v sub 1 points in the positive x direction. Particle 2 has mass m sub 2, is at x=2.0 meters and y=-2.0 meters, and v sub 2 point to the right and down, roughly 45 degrees below the positive x direction. Particle 3 has mass m sub 3, is at x=-3.0 meters and y=1.0 meters, and v sub 3 points down, in the negative y direction. Particle 4has mass m sub 4, is at x=4.0 meters and y=0 meters, and v sub 4 points to the left, in the negative x direction.
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Suppose the particles in the preceding problem have masses m 1 = 0.10 kg, m 2 = 0.20 kg, m 3 = 0.30 kg, m 4 = 0.40 kg . The velocities of the particles are v 1 = 2.0 i ^ m / s , v 2 = ( 3.0 i ^ 3.0 j ^ ) m / s , v 3 = −1.5 j ^ m / s , v 4 = −4.0 i ^ m / s . (a) Calculate the angular momentum of each particle about the origin. (b) What is the total angular momentum of the four-particle system about the origin?

a. l 1 = −0.4 kg · m 2 / s k ^ , l 2 = l 4 = 0 ,
l 3 = 1.35 kg · m 2 / s k ^ ; b. L = 0.95 kg · m 2 / s k ^

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Two particles of equal mass travel with the same speed in opposite directions along parallel lines separated by a distance d . Show that the angular momentum of this two-particle system is the same no matter what point is used as the reference for calculating the angular momentum.

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An airplane of mass 4.0 × 10 4 kg flies horizontally at an altitude of 10 km with a constant speed of 250 m/s relative to Earth. (a) What is the magnitude of the airplane’s angular momentum relative to a ground observer directly below the plane? (b) Does the angular momentum change as the airplane flies along its path?

a. L = 1.0 × 10 11 kg · m 2 / s ; b. No, the angular momentum stays the same since the cross-product involves only the perpendicular distance from the plane to the ground no matter where it is along its path.

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At a particular instant, a 1.0-kg particle’s position is r = ( 2.0 i ^ 4.0 j ^ + 6.0 k ^ ) m , its velocity is v = ( −1.0 i ^ + 4.0 j ^ + 1.0 k ^ ) m / s , and the force on it is F = ( 10.0 i ^ + 15.0 j ^ ) N . (a) What is the angular momentum of the particle about the origin? (b) What is the torque on the particle about the origin? (c) What is the time rate of change of the particle’s angular momentum at this instant?

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A particle of mass m is dropped at the point ( d , 0 ) and falls vertically in Earth’s gravitational field g j ^ . (a) What is the expression for the angular momentum of the particle around the z -axis, which points directly out of the page as shown below? (b) Calculate the torque on the particle around the z -axis. (c) Is the torque equal to the time rate of change of the angular momentum?

An x y coordinate system is shown, with positive x to the right and positive y up. A particle is shown on the x axis, to the left of the y axis, at location minus d comma zero. A force minus m g j hat acts downward on the particle.

a. v = g t j ^ , r = d i ^ , l = m d g t k ^ ;
b. F = m g j ^ , τ = d m g k ^ ; c. yes

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(a) Calculate the angular momentum of Earth in its orbit around the Sun. (b) Compare this angular momentum with the angular momentum of Earth about its axis.

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A boulder of mass 20 kg and radius 20 cm rolls down a hill 15 m high from rest. What is its angular momentum when it is half way down the hill? (b) At the bottom?

a. m g h = 1 2 m ( r ω ) 2 + 1 2 2 5 m r 2 ω 2 ;
ω = 51.2 rad / s ;
L = 16.4 kg · m 2 / s ;
b. ω = 72.5 rad / s ;
L = 23.2 kg · m 2 / s

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A satellite is spinning at 6.0 rev/s. The satellite consists of a main body in the shape of a sphere of radius 2.0 m and mass 10,000 kg, and two antennas projecting out from the center of mass of the main body that can be approximated with rods of length 3.0 m each and mass 10 kg. The antenna’s lie in the plane of rotation. What is the angular momentum of the satellite?

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A propeller consists of two blades each 3.0 m in length and mass 120 kg each. The propeller can be approximated by a single rod rotating about its center of mass. The propeller starts from rest and rotates up to 1200 rpm in 30 seconds at a constant rate. (a) What is the angular momentum of the propeller at t = 10 s; t = 20 s? (b) What is the torque on the propeller?

a. I = 720.0 kg · m 2 ; α = 4.20 rad / s 2 ;
ω ( 10 s ) = 42.0 rad / s ; L = 3.02 × 10 4 kg · m 2 / s ;
ω ( 20 s ) = 84.0 rad / s ;
b. τ = 3.03 × 10 3 N · m

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A pulsar is a rapidly rotating neutron star. The Crab nebula pulsar in the constellation Taurus has a period of 33.5 × 10 −3 s , radius 10.0 km, and mass 2.8 × 10 30 kg . The pulsar’s rotational period will increase over time due to the release of electromagnetic radiation, which doesn’t change its radius but reduces its rotational energy. (a) What is the angular momentum of the pulsar? (b) Suppose the angular velocity decreases at a rate of 10 −14 rad / s 2 . What is the torque on the pulsar?

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The blades of a wind turbine are 30 m in length and rotate at a maximum rotation rate of 20 rev/min. (a) If the blades are 6000 kg each and the rotor assembly has three blades, calculate the angular momentum of the turbine at this rotation rate. (b) What is the torque require to rotate the blades up to the maximum rotation rate in 5 minutes?

a. L = 1.131 × 10 7 kg · m 2 / s ;
b. τ = 3.77 × 10 4 N · m

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A roller coaster has mass 3000.0 kg and needs to make it safely through a vertical circular loop of radius 50.0 m. What is the minimum angular momentum of the coaster at the bottom of the loop to make it safely through? Neglect friction on the track. Take the coaster to be a point particle.

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A mountain biker takes a jump in a race and goes airborne. The mountain bike is travelling at 10.0 m/s before it goes airborne. If the mass of the front wheel on the bike is 750 g and has radius 35 cm, what is the angular momentum of the spinning wheel in the air the moment the bike leaves the ground?

ω = 28.6 rad / s L = 2.6 kg · m 2 / s

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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
An inherent property by virtue of which the body remains in its pure state or initial state
why current is not a vector quantity , whereas it have magnitude as well as direction.
Aniket Reply
the flow of current is not current
bcoz it doesn't satisfy the algabric laws of vectors
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 .
what is binomial theorem
Tollum Reply
hello are you ready to ask aquestion?
Saadaq Reply
what is binary operations
What is the formula to calculat parallel forces that acts in opposite direction?
Martan Reply
position, velocity and acceleration of vector
Manuel Reply
*a plane flies with a velocity of 1000km/hr in a direction North60degree east.find it effective velocity in the easterly and northerly direction.*
hello Lydia.
What is momentum
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
is the eye the same like the camera
I can't understand
same here please
I think the question is that ,,, the working principal of eye and camera same or not?
yes i think is same as the camera
what are the dimensions of surface tension
why is the "_" sign used for a wave to the right instead of to the left?
why classical mechanics is necessary for graduate students?
khyam Reply
classical mechanics?
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
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
2m & m initial velocity 1.8m/s & 4.8m/s respectively,apply conservation of linear momentum in two perpendicular directions.
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
2+5+0=7sec differentiate above equation w.r.t time, as angular velocity is rate of change of angular displacement.
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
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
what is the formula for pressure?
Goodness Reply
force is newtom
and area is meter squared
so in SI units pressure is N/m^2
In customary United States units pressure is lb/in^2. pound per square inch
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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