Rotation Around a Fixed Axis - Wikipedia, The Free Encyclopedia

An example of rotation. Each part of

the worm drive—both the worm and

the worm gear—is rotating on its own

axis.

From Wikipedia, the free encyclopedia

Rotation around a fixed axis is a special case of rotational motion. Thefixed axis hypothesis excludes the possibility of an axis changing itsorientation, and cannot describe such phenomena as wobbling orprecession. According to Euler's rotation theorem, simultaneous rotationalong a number of stationary axes at the same time is impossible. If tworotations are forced at the same time, a new axis of rotation will appear.

This article assumes that the rotation is also stable, such that no torque isrequired to keep it going. The kinematics and dynamics of rotationaround a fixed axis of a rigid body are mathematically much simplerthan those for free rotation of a rigid body; they are entirely analogous tothose of linear motion along a single fixed direction, which is not truefor free rotation of a rigid body. The expressions for the kinetic energyof the object, and for the forces on the parts of the object, are alsosimpler for rotation around a fixed axis, than for general rotationalmotion. For these reasons, rotation around a fixed axis is typically taught in introductory physics courses afterstudents have mastered linear motion; the full generality of rotational motion is not usually taught inintroductory physics classes.

1 Translation and rotation

2 Kinematics

2.1 Angular displacement

2.2 Angular speed and angular velocity

2.3 Angular acceleration

2.4 Equations of kinematics

3 Dynamics

3.1 Moment of inertia

3.2 Torque

3.3 Angular Momentum

3.4 Kinetic energy

4 Vector expression

5 Examples and applications

5.1 Constant angular speed

5.2 Centripetal force

6 See also

7 References

Rotation around a fixed axis - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Rotation_around_a_fixed_axis

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8 Further reading

A rigid body is an object of finite extent in which all the distances between the component particles areconstant. No truly rigid body exists; external forces can deform any solid. For our purposes, then, a rigid body isa solid which requires large forces to deform it appreciably.

A change in the position of a particle in three-dimensional space can be completely specified by threecoordinates. A change in the position of a rigid body is more complicated to describe. It can be regarded as acombination of two distinct types of motion: translational motion and rotational motion.

Purely translational motion occurs when every particle of the body has the same instantaneous velocity as everyother particle; then the path traced out by any particle is exactly parallel to the path traced out by every otherparticle in the body. Under translational motion, the change in the position of a rigid body is specifiedcompletely by three coordinates such as x, y, and z giving the displacement of any point, such as the center ofmass, fixed to the rigid body.

Purely rotational motion occurs if every particle in the body moves in a circle about a single line. This line iscalled the axis of rotation. Then the radius vectors from the axis to all particles undergo the same angulardisplacement in the same time. The axis of rotation need not go through the body. In general, any rotation canbe specified completely by the three angular displacements with respect to the rectangular-coordinate axes x, y,and z. Any change in the position of the rigid body is thus completely described by three translational and threerotational coordinates.

Any displacement of a rigid body may be arrived at by first subjecting the body to a displacement followed by arotation, or conversely, to a rotation followed by a displacement. We already know that for any collection ofparticles—whether at rest with respect to one another, as in a rigid body, or in relative motion, like theexploding fragments of a shell, the acceleration of the center of mass is given by

where M is the total mass of the system and acm is the acceleration of the center of mass. There remains the

matter of describing the rotation of the body about the center of mass and relating it to the external forces actingon the body. The kinematics and dynamics of rotational motion around a single axis resemble the kinematicsand dynamics of translational motion; rotational motion around a single axis even has a work-energy theoremanalogous to that of particle dynamics.

Angular displacement

A particle moves in a circle of radius . Having moved an arc length , its angular position is relative to its

original position, where .

In mathematics and physics it is usual to use the natural unit radians rather than degrees or revolutions. Units


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are converted as follows:

[1]

An angular displacement is a change in angular position:

where is the angular displacement, is the initial angular position and is the final angular position.[1]

Angular speed and angular velocity

Angular velocity is the change in angular displacement per unit time. The symbol for angular velocity is and

the units are typically rad s−1. Angular speed is the magnitude of angular velocity.

The instantaneous angular velocity is given by

Using the formula for angular position and letting , we have also

where is the translational speed of the particle.

Angular velocity and frequency are related by

.

Angular acceleration

A changing angular velocity indicates the presence of an angular acceleration in rigid body, typically measured

in rad s−2. The average angular acceleration over a time interval Δt is given by

The instantaneous acceleration α(t) is given by


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Thus, the angular acceleration is the rate of change of the angular velocity, just as acceleration is the rate ofchange of velocity.

The translational acceleration of a point on the object rotating is given by

where r is the radius or distance from the axis of rotation. This is also the tangential component of acceleration:it is tangential to the direction of motion of the point. If this component is 0, the motion is uniform circularmotion, and the velocity changes in direction only.

The radial acceleration (perpendicular to direction of motion) is given by

.

It is directed towards the center of the rotational motion, and is often called the centripetal acceleration.

The angular acceleration is caused by the torque, which can have a positive or negative value in accordancewith the convention of positive and negative angular frequency. The ratio of torque and angular acceleration(how difficult it is to start, stop, or otherwise change rotation) is given by the moment of inertia: .

Equations of kinematics

When the angular acceleration is constant, the five quantities angular displacement , initial angular velocity , final angular velocity , angular acceleration , and time can be related by four equations of kinematics:

Moment of inertia

The moment of inertia of an object, symbolized by I, is a measure of the object's resistance to changes to itsrotation. The moment of inertia is measured in kilogram metre² (kg m²). It depends on the object's mass:increasing the mass of an object increases the moment of inertia. It also depends on the distribution of the mass:distributing the mass further from the centre of rotation increases the moment of inertia by a greater degree. Fora single particle of mass a distance from the axis of rotation, the moment of inertia is given by


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Torque

Torque is the twisting effect of a force F applied to a rotating object which is at position r from its axis ofrotation. Mathematically,

where × denotes the cross product. A net torque acting upon an object will produce an angular acceleration ofthe object according to

just as F = ma in linear dynamics.

The work done by a torque acting on an object equals the magnitude of the torque times the angle throughwhich the torque is applied:

The power of a torque is equal to the work done by the torque per unit time, hence:

Angular Momentum

The angular momentum L is a measure of the difficulty of bringing a rotating object to rest. It is given by

Angular momentum is related to angular velocity by

just as p = mv in linear dynamics.

The equivalent of linear momentum in rotational motion is angular momentum. The greater the angularmomentum of the spinning object such as a top, the greater its tendency to continue to spin.

The Angular Momentum of a rotating body is proportional to its mass and to how rapidly it is turning. Inaddition the angular momentum depends on how the mass is distributed relative to the axis of rotation: thefurther away the mass is located from the axis of rotation, the greater the angular momentum . A flat disk suchas a record turntable has less angular momentum than a hollow cylinder of the same mass and velocity ofrotation.

Like linear momentum, angular momentum is vector quantity, and its conservation implies that the direction ofthe spin axis tends to remain unchanged. For this reason the spinning top remains upright whereas a stationaryone falls over immediately.

The angular momentum equation can be used to relate the moment of the resultant force on a body about an axis(sometimes called torque), and the rate of rotation about that axis.


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Torque and angular momentum are related according to

just as F = dp/dt in linear dynamics. In the absence of an external torque, the angular momentum of a bodyremains constant. The conservation of angular momentum is notably demonstrated in figure skating: whenpulling the arms closer to the body during a spin, the moment of inertia is decreased, and so the angular velocityis increased.

Kinetic energy

The kinetic energy Krot due to the rotation of the body is given by

just as Ktrans = 1⁄2mv2 in linear dynamics.

The above development is a special case of general rotational motion. In the general case, angular displacement,angular velocity, angular acceleration and torque are considered to be vectors.

An angular displacement is considered to be a vector, pointing along the axis, of magnitude equal to that of .A right-hand rule is used to find which way it points along the axis; if the fingers of the right hand are curled topoint in the way that the object has rotated, then the thumb of the right hand points in the direction of the vector.

The angular velocity vector also points along the axis of rotation in the same way as the angular displacementsit causes. If a disk spins counterclockwise as seen from above, its angular velocity vector points upwards.Similarly, the angular acceleration vector points along the axis of rotation in the same direction that the angularvelocity would point if the angular acceleration were maintained for a long time.

The torque vector points along the axis around which the torque tends to cause rotation. To maintain rotationaround a fixed axis, the total torque vector has to be along the axis, so that it only changes the magnitude andnot the direction of the angular velocity vector. In the case of a hinge, only the component of the torque vectoralong the axis has effect on the rotation, other forces and torques are compensated by the structure.

Constant angular speed

The simplest case of rotation around a fixed axis is that of constant angular speed. Then the total torque is zero.For the example of the Earth rotating around its axis, there is very little friction. For a fan, the motor applies atorque to compensate for friction. The angle of rotation is a linear function of time, which modulo 360° is aperiodic function.

An example of this is the two-body problem with circular orbits.


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Centripetal force

Internal tensile stress provides the centripetal force that keeps a spinning object together. A rigid body modelneglects the accompanying strain. If the body is not rigid this strain will cause it to change shape. This isexpressed as the object changing shape due to the "centrifugal force".

Celestial bodies rotating about each other often have elliptic orbits. The special case of circular orbits is anexample of a rotation around a fixed axis: this axis is the line through the center of mass perpendicular to theplane of motion. The centripetal force is provided by gravity, see also two-body problem. This usually alsoapplies for a spinning celestial body, so it need not be solid to keep together, unless the angular speed is too highin relation to its density. (It will, however, tend to become oblate.) For example, a spinning celestial body ofwater must take at least 3 hours and 18 minutes to rotate, regardless of size, or the water will separate. If thedensity of the fluid is higher the time can be less. See orbital period.

Anatomical terms of motion

Artificial gravity by rotation

Axle

Carousel, Ferris wheel

Centrifugal force

Centrifuge

Centripetal force

Circular motion

Coriolis effect

Fictitious force

Flywheel

Gyration

Linear-rotational analogs

Revolutions per minute

Revolving door

Rigid body angular momentum

Rotational speed

Rotational symmetry

Spin

^ a b Fundamentals of Physics Extended 7th Edition by Halliday, Resnick and Walker. ISBN 0-471-23231-91.

Concepts of Physics Volume 1, 1st edition Seventh reprint by Harish Chandra Verma ISBN 81-7709-187-5

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Categories: Celestial mechanics Euclidean symmetries Rotation

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