Circular Motion - Wikipedia, The Free Encyclopedia

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Circular motionFrom Wikipedia, the free encyclopedia

In physics, circular motion is a movement of an object along the circumference of a circle or rotation along

a circular path. It can be uniform, with constant angular rate of rotation (and constant speed), or non-uniform

with a changing rate of rotation. The rotation around a fixed axis of a three-dimensional body involves

circular motion of its parts. The equations of motion describe the movement of the center of mass of a body.

Examples of circular motion include: an artificial satellite orbiting the Earth at constant height, a stone

which is tied to a rope and is being swung in circles, a car turning through a curve in a race track, an electron

moving perpendicular to a uniform magnetic field, and a gear turning inside a mechanism.

Since the object's velocity vector is constantly changing direction, the moving object is undergoing

acceleration by a centripetal force in the direction of the center of rotation. Without this acceleration, the

object would move in a straight line, according to Newton's laws of motion.

Contents

1 Uniform circular motion1.1 Formulas for uniform circular motion1.2 In polar coordinates

1.3 Using complex numbers2 Velocity3 Acceleration

4 Non-uniform5 Applications6 References7 External links8 See also

Uniform circular motion

In physics, uniform circular motion describes the motion of a body traversing a circular path at constantspeed. The distance of the body from the axis of rotation remains constant at all times. Though the body's

speed is constant, its velocity is not constant: velocity, a vector quantity, depends on both the body's speed

and its direction of travel. This changing velocity indicates the presence of an acceleration; this centripetal

acceleration is of constant magnitude and directed at all times towards the axis of rotation. This acceleration

is, in turn, produced by a centripetal force which is also constant in magnitude and directed towards the axis

of rotation.

In the case of rotation around a fixed axis of a rigid body that is not negligibly small compared to the radius

of the path, each particle of the body describes a uniform circular motion with the same angular velocity, but

with velocity and acceleration varying with the position with respect to the axis.
http://en.wikipedia.org/wiki/Centripetal_forcehttp://en.wikipedia.org/wiki/Centripetal_accelerationhttp://en.wikipedia.org/wiki/Velocityhttp://en.wikipedia.org/wiki/Euclidean_vectorhttp://en.wikipedia.org/wiki/Physicshttp://en.wikipedia.org/wiki/Circlehttp://en.wikipedia.org/wiki/Circular_motion#External_linkshttp://en.wikipedia.org/wiki/Circular_motion#Non-uniformhttp://en.wikipedia.org/wiki/Circular_motion#Accelerationhttp://en.wikipedia.org/wiki/Circular_motion#Using_complex_numbershttp://en.wikipedia.org/wiki/Circular_motion#In_polar_coordinateshttp://en.wikipedia.org/wiki/Circular_motion#Formulas_for_uniform_circular_motionhttp://en.wikipedia.org/wiki/Circular_motion#Uniform_circular_motionhttp://en.wikipedia.org/wiki/Center_of_masshttp://en.wikipedia.org/wiki/Rotation_around_a_fixed_axishttp://en.wikipedia.org/wiki/Rigid_bodyhttp://en.wikipedia.org/wiki/Rotation_around_a_fixed_axishttp://en.wikipedia.org/wiki/Centripetal_forcehttp://en.wikipedia.org/wiki/Centripetal_accelerationhttp://en.wikipedia.org/wiki/Euclidean_vectorhttp://en.wikipedia.org/wiki/Velocityhttp://en.wikipedia.org/wiki/Distancehttp://en.wikipedia.org/wiki/Speedhttp://en.wikipedia.org/wiki/Circlehttp://en.wikipedia.org/wiki/Physicshttp://en.wikipedia.org/wiki/Circular_motion#See_alsohttp://en.wikipedia.org/wiki/Circular_motion#External_linkshttp://en.wikipedia.org/wiki/Circular_motion#Referenceshttp://en.wikipedia.org/wiki/Circular_motion#Applicationshttp://en.wikipedia.org/wiki/Circular_motion#Non-uniformhttp://en.wikipedia.org/wiki/Circular_motion#Accelerationhttp://en.wikipedia.org/wiki/Circular_motion#Velocityhttp://en.wikipedia.org/wiki/Circular_motion#Using_complex_numbershttp://en.wikipedia.org/wiki/Circular_motion#In_polar_coordinateshttp://en.wikipedia.org/wiki/Circular_motion#Formulas_for_uniform_circular_motionhttp://en.wikipedia.org/wiki/Circular_motion#Uniform_circular_motionhttp://en.wikipedia.org/wiki/Newton%27s_laws_of_motionhttp://en.wikipedia.org/wiki/Centripetal_forcehttp://en.wikipedia.org/wiki/Accelerationhttp://en.wikipedia.org/wiki/Gearhttp://en.wikipedia.org/wiki/Magnetic_fieldhttp://en.wikipedia.org/wiki/Race_trackhttp://en.wikipedia.org/wiki/Center_of_masshttp://en.wikipedia.org/wiki/Rotation_around_a_fixed_axishttp://en.wikipedia.org/wiki/Rotationhttp://en.wikipedia.org/wiki/Circlehttp://en.wikipedia.org/wiki/Circumferencehttp://en.wikipedia.org/wiki/Physics


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Figure 1: Velocity v and

acceleration a in uniform

circular motion at angular rate

; the speed is constant, but

the velocity is always tangent

to the orbit; the acceleration

has constant magnitude, but

always points toward the

center of rotation

Figure 2: The velocity vectors at time tand

time t+ dtare moved from the orbit on the

left to new positions where their tails

coincide, on the right. Because the velocity

is fixed in magnitude at v = r, the

velocity vectors also sweep out a circular

path at angular rate . As dt 0, the

acceleration vector a becomes

perpendicular to v, which means it points

toward the center of the orbit in the circle

on the left. Angle dtis the very small

angle between the two velocities and tends

to zero as dt 0

Formulas for uniform circular motion

For motion in a circle of radius r, the circumference of the circle is C= 2

r. If the period for one rotation is T, the angular rate of rotation, also known

as angular velocity, is:

and the units are radians/sec

The speed of the object traveling the circle is:

The angle swept out in a time tis:

The acceleration due to change in the direction is:

The vector relationships are shown in Figure 1. The axis of

rotation is shown as a vector perpendicular to the plane of the

orbit and with a magnitude = d / dt. The direction of is

chosen using the right-hand rule. With this convention for

depicting rotation, the velocity is given by a vector cross product

as

which is a vector perpendicular to both and r ( t), tangential

to the orbit, and of magnitude r. Likewise, the acceleration is

given by

which is a vector perpendicular to both and v ( t) of

magnitude |v| = 2r and directed exactly opposite to r ( t).[1]

In the simplest case the speed, mass and radius are constant.

Consider a body of one kilogram, moving in a circle of radius

one metre, with an angular velocity of one radian per second.

The speed is one metre per second.The inward acceleration is one metre per square second[v^2/r]
http://en.wikipedia.org/wiki/Accelerationhttp://en.wikipedia.org/wiki/Speedhttp://en.wikipedia.org/wiki/Secondhttp://en.wikipedia.org/wiki/Radianhttp://en.wikipedia.org/wiki/Angular_velocityhttp://en.wikipedia.org/wiki/Radiushttp://en.wikipedia.org/wiki/Circular_motion#cite_note-1http://en.wikipedia.org/wiki/Cross_producthttp://en.wikipedia.org/wiki/Right-hand_rulehttp://en.wikipedia.org/wiki/Angular_velocityhttp://en.wikipedia.org/wiki/Radiushttp://en.wikipedia.org/wiki/File:Velocity-acceleration.PNGhttp://en.wikipedia.org/wiki/File:Uniform_circular_motion.svg


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Figure 3: (Left) Ball in circular motion rope

provides centripetal force to keep ball in circle

(Right) Rope is cut and ball continues in straight

line with velocity at the time of cutting the rope, inaccord with Newton's law of inertia, because

centripetal force is no longer there

Figure 1: Vector relationships for uniform circular

motion; vector representing the rotation is

normal to the plane of the orbit.

It is subject to a centripetal force of one kilogram metre per square second, which is one newton.

The momentum of the body is one kgms1.

The moment of inertia is one kgm2.

The angular momentum is one kgm2s1.The kinetic energy is 1/2 joule.

The circumference of the orbit is 2 (~ 6.283)

metres.The period of the motion is 2seconds per turn.

The frequency is (2)1 hertz.

In polar coordinates

During circular motion the body moves on a curve that

can be described in polar coordinate system as a fixed

distanceR from the center of the orbit taken as origin,

oriented at an angle (t) from some reference direction.

See Figure 2. The displacement vector is the radialvector from the origin to the particle location:

where is the unit vector parallel to the radius

vector at time tand pointing away from the origin. It is

convenient to introduce the unit vector orthogonal to

as well, namely . It is customary to orient to point

in the direction of travel along the orbit.

The velocity is the time derivative of the displacement:

Because the radius of the circle is constant, the radial

component of the velocity is zero. The unit vector has

a time-invariant magnitude of unity, so as time varies its

tip always lies on a circle of unit radius, with an angle the same as the angle of . If the particle

displacement rotates through an angle d in time dt, so does , describing an arc on the unit circle of

magnitude d. See the unit circle at the left of Figure 2. Hence:

where the direction of the change must be perpendicular to (or, in other words, along ) because any

change d in the direction of would change the size of . The sign is positive, because an increase in

d implies the object and have moved in the direction of . Hence the velocity becomes:
http://en.wikipedia.org/wiki/Orthogonality#Euclidean_vector_spaceshttp://en.wikipedia.org/wiki/Unit_vectorhttp://en.wikipedia.org/wiki/Polar_coordinate_systemhttp://en.wikipedia.org/wiki/Hertzhttp://en.wikipedia.org/wiki/Frequencyhttp://en.wikipedia.org/wiki/Turn_(geometry)http://en.wikipedia.org/wiki/Pihttp://en.wikipedia.org/wiki/Orbithttp://en.wikipedia.org/wiki/Circumferencehttp://en.wikipedia.org/wiki/Joulehttp://en.wikipedia.org/wiki/Kinetic_energyhttp://en.wikipedia.org/wiki/Angular_momentumhttp://en.wikipedia.org/wiki/Moment_of_inertiahttp://en.wikipedia.org/wiki/Momentumhttp://en.wikipedia.org/wiki/Newton_(unit)http://en.wikipedia.org/wiki/Centripetal_forcehttp://en.wikipedia.org/wiki/File:Circular_motion_vectors.svghttp://en.wikipedia.org/wiki/File:Breaking_String.PNG


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Figure 2: Polar coordinates for circular trajectory. On the left

is a unit circle showing the changes and in the

unit vectors and for a small increment in angle

.

The acceleration of the body can also be broken into radial and tangential components. The acceleration is

the time derivative of the velocity:

The time derivative of is found the same way as for . Again, is a unit vector and its tip traces a unit

circle with an angle that is /2 + . Hence, an increase in angle d by implies traces an arc of

magnitude d, and as is orthogonal to , we have:

where a negative sign is necessary to keep

orthogonal to . (Otherwise, the angle between

and would decrease with increase in d.)

See the unit circle at the left of Figure 2.

Consequently the acceleration is:

The centripetal acceleration is the radial

component, which is directed radially inward:

while the tangential component changes the magnitude of the velocity:

Using complex numbers

Circular motion can be described using complex numbers. Let the axis be the real axis and the axis be

the imaginary axis. The position of the body can then be given as , a complex "vector":

where is the imaginary unit, and
http://en.wikipedia.org/wiki/Imaginary_unithttp://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Vector_(geometry)#Lengthhttp://en.wikipedia.org/wiki/Centripetal_forcehttp://en.wikipedia.org/wiki/File:Vectors_in_polar_coordinates.PNG


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is the angle of the complex vector with the real axis and is a function of time t. Since the radius is constant:

where a dotindicates time differentiation. With this notation the velocity becomes:

and the acceleration becomes:

The first term is opposite in direction to the displacement vector and the second is perpendicular to it, just

like the earlier results shown before.

Velocity

Figure 1 illustrates velocity and acceleration vectors for uniform motion at four different points in the orbit.

Because the velocity v is tangent to the circular path, no two velocities point in the same direction. Although

the object has a constant speed, its direction is always changing. This change in velocity is caused by anacceleration a, whose magnitude is (like that of the velocity) held constant, but whose direction also is

always changing. The acceleration points radially inwards (centripetally) and is perpendicular to the

velocity. This acceleration is known as centripetal acceleration.

For a path of radius r, when an angle is swept out, the distance travelled on the periphery of the orbit is s =

r. Therefore, the speed of travel around the orbit is

,

where the angular rate of rotation is . (By rearrangement, = v/r.) Thus, v is a constant, and the velocity

vector v also rotates with constant magnitude v, at the same angular rate .

Acceleration

Main article: Acceleration

The left-hand circle in Figure 2 is the orbit showing the velocity vectors at two adjacent times. On the right,

these two velocities are moved so their tails coincide. Because speed is constant, the velocity vectors on the

right sweep out a circle as time advances. For a swept angle d = dtthe change in v is a vector at right

angles to v and of magnitude vd, which in turn means that the magnitude of the acceleration is given by
http://en.wikipedia.org/wiki/Accelerationhttp://en.wiktionary.org/wiki/peripheryhttp://en.wikipedia.org/wiki/Centripetalhttp://en.wikipedia.org/wiki/Acceleration


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Centripetal acceleration for some values of radius and magnitude of velocity

|v|

r

1 m/s3.6 km/h

2.2 mph

2 m/s7.2 km/h

4.5 mph

5 m/s18 km/h

11 mph

10 m/s36 km/h

22 mph

20 m/s72 km/h

45 mph

50 m/s180 km/h

110 mph

100 m/s360 km/h

220 mphSlow walk Bicycle City car Aerobatics

10 cm3.9 in

Laboratorycentrifuge

10 m/s

1.0 g40 m/s

4.1 g250 m/s

25 g1.0 km/s

100 g4.0 km/s

410 g25 km/s

2500 g100 km/s

10000 g

20 cm7.9 in

5.0 m/s

0.51 g20 m/s

2.0 g130 m/s

13 g500 m/s

51 g2.0 km/s

200 g13 km/s

1300 g50 km/s

5100 g

50 cm1.6 ft

2.0 m/s

0.20 g8.0 m/s

0.82 g50 m/s

5.1 g200 m/s

20 g800 m/s

82 g5.0 km/s

510 g20 km/s

2000 g

1 m

3.3 ft

Playground

carousel

1.0 m/s

0.10 g

4.0 m/s

0.41 g

25 m/s

2.5 g

100 m/s

10 g

400 m/s

41 g

2.5 km/s

250 g

10 km/s

1000 g

2 m6.6 ft

500 mm/s

0.051 g2.0 m/s

0.20 g13 m/s

1.3 g50 m/s

5.1 g200 m/s

20 g1.3 km/s

130 g5.0 km/s

510 g

5 m16 ft

200 mm/s

0.020 g800 mm/s

0.082 g5.0 m/s

0.51 g20 m/s

2.0 g80 m/s

8.2 g500 m/s

51 g2.0 km/s

200 g

10 m33 ft

Roller-coastervertical loop

100 mm/s

0.010 g400 mm/s

0.041 g2.5 m/s

0.25 g10 m/s

1.0 g40 m/s

4.1 g250 m/s

25 g1.0 km/s

100 g

20 m66 ft

50 mm/s

0.0051g

200 mm/s

0.020g

1.3 m/s

0.13g

5.0 m/s

0.51g

20 m/s

2g

130 m/s

13g

500 m/s

51g

50 m160 ft

20 mm/s

0.0020 g80 mm/s

0.0082 g500 mm/s

0.051 g2.0 m/s

0.20 g8.0 m/s

0.82 g50 m/s

5.1 g200 m/s

20 g

100 m330 ft

Freewayon-ramp

10 mm/s0.0010 g

40 mm/s0.0041 g

250 mm/s0.025 g

1.0 m/s0.10 g

4.0 m/s0.41 g

25 m/s2.5 g

100 m/s10 g

200 m660 ft

5.0 mm/s0.00051 g

20 mm/s0.0020 g

130 m/s0.013 g

500 mm/s0.051 g

2.0 m/s0.20 g

13 m/s1.3 g

50 m/s5.1 g

500 m1600 ft

2.0 mm/s0.00020 g

8.0 mm/s0.00082 g

50 mm/s0.0051 g

200 mm/s0.020 g

800 mm/s0.082 g

5.0 m/s0.51 g

20 m/s2.0 g

1 km3300 ft

High-speedrailway

1.0 mm/s0.00010 g

4.0 mm/s0.00041 g

25 mm/s0.0025 g

100 mm/s0.010 g

400 mm/s0.041 g

2.5 m/s0.25 g

10 m/s1.0 g

Non-uniform

Non-uniform circular motion is any case in which an object moving in a circular path has a varying speed.

The tangential acceleration is non-zero; the speed is changing.

Since there is a non-zero tangential acceleration, there are forces that act on an object in addition to its

centripetal force (composed of the mass and radial acceleration). These forces include weight, normal force,and friction.
http://en.wikipedia.org/wiki/Frictionhttp://en.wikipedia.org/wiki/Normal_forcehttp://en.wikipedia.org/wiki/Centripetal_forcehttp://en.wikipedia.org/wiki/Tangential_accelerationhttp://en.wikipedia.org/wiki/Speedhttp://en.wikipedia.org/wiki/High-speed_railhttp://en.wikipedia.org/wiki/Entrance_ramphttp://en.wikipedia.org/wiki/Freewayhttp://en.wikipedia.org/wiki/Vertical_loophttp://en.wikipedia.org/wiki/Roller-coasterhttp://en.wikipedia.org/wiki/Carouselhttp://en.wikipedia.org/wiki/Playgroundhttp://en.wikipedia.org/wiki/Laboratory_centrifugehttp://en.wikipedia.org/wiki/Aerobaticshttp://en.wikipedia.org/wiki/City_carhttp://en.wikipedia.org/wiki/Bicyclehttp://en.wikipedia.org/wiki/Walk


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In non-uniform circular motion, normal force does not always point in the opposite direction of weight. Here

is an example with an object traveling in a straight path then loops a loop back into a straight path again.

This diagram shows the normal force pointing in other directions rather than opposite to the weight force.

The normal force is actually the sum of the radial and tangential forces that help to counteract the weight

force and contribute to the centripetal force. The horizontal component of normal force is what contributes

to the centripetal force. The vertical component of the normal force is what counteracts the weight of the

object.

In non-uniform circular motion, normal force and weight

may point in the same direction. Both forces can point

down, yet the object will remain in a circular path without

falling straight down. First lets see why normal force can

point down in the first place. In the first diagram, let's say

the object is a person sitting inside a plane, the two forces

point down only when it reaches the top of the circle. The

reason for this is that the normal force is the sum of the

weight and centripetal force. Since both weight and

centripetal force points down at the top of the circle, normal force

will point down as well. From a logical standpoint, a person who is

traveling in the plane will be upside down at the top of the circle. At

that moment, the persons seat is actually pushing down on the

person, which is the normal force.

The reason why the object does not fall down when subjected to only

downward forces is a simple one. Think about what keeps an object

up after it is thrown. Once an object is thrown into the air, there is

only the downward force of earths gravity that acts on the object.

That does not mean that once an object is thrown in the air, it will

fall instantly. What keeps that object up in the air is its velocity. The

first of Newton's laws of motion states that an objects inertia keeps

it in motion, and since the object in the air has a velocity, it will tend

to keep moving in that direction.

Applications

Solving applications dealing with non-uniform circular motioninvolves force analysis. With uniform circular motion, the only force

acting upon an object traveling in a circle is the centripetal force. In

non-uniform circular motion, there are additional forces acting on the

object due to a non-zero tangential acceleration. Although there are

additional forces acting upon the object, the sum of all the forces

acting on the object will have to equal to the centripetal force.
http://en.wikipedia.org/wiki/Inertiahttp://en.wikipedia.org/wiki/Newton%27s_laws_of_motionhttp://en.wikipedia.org/wiki/Velocityhttp://en.wikipedia.org/wiki/Weighthttp://en.wikipedia.org/wiki/File:Normal_and_weight.svghttp://en.wikipedia.org/wiki/File:Freebody_object.svghttp://en.wikipedia.org/wiki/File:Freebody_circular.svghttp://en.wikipedia.org/wiki/File:Nonuniform_circular_motion.svg


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Radial acceleration is used when calculating the total force. Tangential acceleration is not used in calculating

total force because it is not responsible for keeping the object in a circular path. The only acceleration

responsible for keeping an object moving in a circle is the radial acceleration. Since the sum of all forces isthe centripetal force, drawing centripetal force into a free body diagram is not necessary and usually not

recommended.

Using , we can draw free body diagrams to list all the forces acting on an object then set it

equal to . Afterwards, we can solve for what ever is unknown (this can be mass, velocity, radius of

curvature, coefficient of friction, normal force, etc.). For example, the visual above showing an object at the

top of a semicircle would be expressed as .

In uniform circular motion, total acceleration of an object in a circular path is equal to the radial

acceleration. Due to the presence of tangential acceleration in non uniform circular motion, that does nothold true any more. To find the total acceleration of an object in non uniform circular, find the vector sum of

the tangential acceleration and the radial acceleration.

Radial acceleration is still equal to . Tangential acceleration is simply the derivative of the velocity at

any given point: . This root sum of squares of separate radial and tangential accelerations is

only correct for circular motion; for general motion within a plane with polar coordinates , the Coriolis

term should be added to , whereas radial acceleration then becomes.

References

1. ^ Knudsen, Jens M.; Hjorth, Poul G. (2000).Elements of Newtonian mechanics: including nonlinear dynamics

(http://books.google.com/books?id=Urumwws_lWUC) (3 ed.). Springer. p. 96. ISBN 3-540-67652-X., Chapter 5

page 96 (http://books.google.com/books?id=Urumwws_lWUC&pg=PA96)

External links

Physclips: Mechanics with animations and video clips (http://www.physclips.unsw.edu.au/) from theUniversity of New South WalesCircular Motion (http://www.lightandmatter.com/html_books/1np/ch09/ch09.html) a chapter froman online textbook

Circular Motion Lecture (http://ocw.mit.edu/OcwWeb/Physics/8-01Physics-IFall1999/VideoLectures/detail/embed05.htm) a video lecture on CM

See alsoAngular momentum
http://en.wikipedia.org/wiki/Angular_momentumhttp://ocw.mit.edu/OcwWeb/Physics/8-01Physics-IFall1999/VideoLectures/detail/embed05.htmhttp://www.lightandmatter.com/html_books/1np/ch09/ch09.htmlhttp://www.physclips.unsw.edu.au/http://books.google.com/books?id=Urumwws_lWUC&pg=PA96http://en.wikipedia.org/wiki/Special:BookSources/3-540-67652-Xhttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://books.google.com/books?id=Urumwws_lWUChttp://en.wikipedia.org/wiki/Circular_motion#cite_ref-1


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Equations of motion for circular motionPendulum (mathematics)Reciprocating motion

Simple harmonic motionExample: circular motionGeostationary orbitFictitious force

Geosynchronous orbitExample: Circular motionReactive centrifugal forceSling (weapon)

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