45871255 Rotational Motion

8/14/2019 45871255 Rotational Motion

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Lab. 3 Rotational MotionMost fundamental concepts are substracted from the web site:

http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html

Circular Motionhttp://hyperphysics.phy-astr.gsu.edu/hbase/circ.html#rotcon

For circular motion at a constant speed v, the centripetal acceleration of the motion can be derived.

Since in radian measure ,

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Basic Rotational Quantities

(n addition to any tangential acceleration, there is always the

centripetal acceleration :

)he angular displancment is defined by:

For a circular path it follows that the

angular velocity is

and the angular acceleration is

where the acceleration here is thetangential acceleration.

)he standard angle of a directed *uantity is ta+en to be countercloc+wise from the positive a is.

or!uehttp://hyperphysics.phy-astr.gsu.edu/hbase/tor!.html#tor!

tor*ue is an influence which tends to change the rotational motion of an ob ect. ne way to

*uantify a tor*ue is

or!ue " orce applied $ le%er arm

)he lever arm is defined as the perpendicular distance from the a is of rotation to the line of action

of the force.

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Right &and Rule 'or or!ue

)or*ue is inherently a vector *uantity . %art of the tor*ue calculation is the determination of

direction. )he direction is perpendicular to both the radius from the a is and to the force. (t is

conventional to choose it in the right hand rule direction along the a is of rotation. )he tor*ue is in

the direction of the angular velocity which would be produced by it in the absence of other

influences. (n general, the change in angular velocity is in the direction of the tor*ue.

(e)ton*s +econd La)

0ewton1s Second 2aw as stated below applies to a wide range of physical phenomena, but it is not a

fundamental principle li+e the 3onservation 2aws . (t is applicable only if the force is the net

e ternal force. (t does not apply directly to situations where the mass is changing, either from loss

or gain of material, or because the ob ect is traveling close to the speed of light where relativistic

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effects must be included. (t does not apply directly on the very small scale of the atom where

*uantum mechanics must be used.

ata can be entered into any of the bo es below. Specifying any two of the *uantities determines

the third. fter you have entered values for two, clic+ on the te t representing to third to calculate

its value.

0ewtons 5 +g 6 m/s '

pounds 5 slugs 6 ft/s '

2imitations of 0ewton1s Second 2aw

(e)ton*s +econd La) ,llustration

0ewton1s 'nd 2aw enables us to compare the results of the same force e erted on ob ects of

different mass.

Limitations on (e)ton*s nd La)

ne of the best +nown relationships in physics is 0ewton1s 'nd 2aw

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but, though e tremely useful, it is not a fundamental principle li+e the conservation laws. F must be

the net e ternal force, and even then a more fundamental relationship is

)he net force should be defined as the rate of change of momentum 7 this becomes

only if the mass is constant. Since the mass changes as the speed approaches the speed of light ,

F5ma is seen to be strictly a non-relativistic relationship which applies to the acceleration of

constant mass ob ects. espite these limitations, it is e tremely useful for the prediction of motion

under these constraints.

ariable Mass pplications

)he generali8ation of 0ewton1s 'nd 2aw to apply to variable mass systems ta+es the form

)he term involving the derivative of the mass is responsible for the thrust in roc+et propulsion and

must be included in any problem where the mass changes.

(e)ton*s nd La): Rotationhttp://hyperphysics.phy-astr.gsu.edu/hbase/h'rame.html

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)he relationship between the net e ternal tor*ue and the angular acceleration is of the same form as

0ewton1s second law and is sometimes called 0ewton1s second law for rotation. (t is not as general

a relationship as the linear one because the moment of inertia is not strictly a scalar *uantity. )he

rotational e*uation is limited to rotation about a single principal a is , which in simple cases is an

a is of symmetry.

(et e$ternal tor!ue " moment o' inertia $ angular acceleration

0 m 5 +g m ' rad/s '

0 m 5 +g m ' rad/s '

9ou may enter data for any two of the *uantities and then clic+ on the active te t for the *uantity

you wish to calculate. )he data values will not be forced to be consistent until you clic+ on the

*uantity to calculate.

Conditions 'or 0!uilibrium

n ob ect at e*uilibrium has no net influences to cause it to move, either in translation linear

motion; or rotation. )he basic conditions for e*uilibrium are:

)he conditions for e*uilibrium are basic to the design of any load-bearing structure such as a bridge

or a building since such structures must be able to maintain e*uilibrium under load. )hey are alsoimportant for the study of machines, since one must first establish e*uilibrium and then apply e tra

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force or tor*ue to produce the desired movement of the machine. )he conditions of e*uilibrium are

used to analy8e the < simple machines < which are the building bloc+s for more comple machines.

Moment o' ,nertiaMoment of inertia is the name given to rotational inertia, the rotational analog of mass for linear

motion. (t appears in the relationships for the dynamics of rotational motion. )he moment of inertia

must be specified with respect to a chosen a is of rotation. For a point mass the moment of inertia is

ust the mass times the s*uare of perpendicular distance to the rotation a is, ( 5 mr ' . )hat point

mass relationship becomes the basis for all other moments of inertia since any ob ect can be built up

from a collection of point masses.

1arallel $is heoremhttp://hyperphysics.phy-astr.gsu.edu/hbase/para$.html

)he moment of inertia of any ob ect about an a is through its

center of mass is the minimum moment of inertia for an a is

in that direction in space. )he moment of inertia about any

a is parallel to that a is through the center of mass is given by

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1erpendicular $is heoremFor a planar ob ect, the moment of inertia about an a is perpendicular to the plane is the sum of the

moments of inertia of two perpendicular a es through the same point in the plane of the ob ect. )he

utility of this theorem goes beyond that of calculating moments of strictly planar ob ects. (t is a

valuable tool in the building up of the moments of inertia of three dimensional ob ects such as

cylinders by brea+ing them up into planar dis+s and summing the moments of inertia of the

composite dis+s .

Show the development of the relationship.

1erpendicular $is heorem)he perpendicular a is theorem for planar ob ects can be demonstrated by loo+ing at the contribution

to the three a is moments of inertia from an arbitrary mass element. From the point mass moment,

the contributions to each of the a is moments of inertia are

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Rotational and Linear Example A mass m is placed on a rod of length r and negligible mass, and constrained to rotateabout a fixed axis. If the mass is released from a horizontal orientation, it can be describedeither in terms of force and accleration with Newton's second law for linear motion, or as apure rotation about the axis with Newton's second law for rotation . This provides a settingfor comparing linear and rotational quantities for the same s stem. This process leads tothe expression for the moment of inertia of a point mass.

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Moment o' ,nertia: RodFor a uniform rod with negligible thic+ness, the moment of inertia about its center ofmass is

For mass M 5 +g and length

2 5 m, the moment of inertia is

( 5 +g m? Show

)he moment of inertia about the end of the rod can be calculated directly or obtained from the center

of mass e pression by use of the %arallel a is theorem .

)he moment of inertia about the end of the rod is

( 5 +g m?. Show

,' the thic2ness is not negligible then the e$pression 'or , o' a cylinder about itsend can be used.

Moment o' ,nertia: Rod3alculating the moment of inertia of a rod about its center of mass is a good e ample of the need for

calculus to deal with the properties of continuous mass distributions. )he moment of inertia of a pointmass is given by ( 5 mr ' , but the rod would have to be considered to be an infinite number of point

masses, and each must be multiplied by the s*uare of its distance from the a is. )he resulting infinite

sum is called an integral . )he general form for the moment of inertia is:

@hen the mass element dm is e pressed in terms of a length element dr along the rod and the sum

ta+en over the entire length, the integral ta+es the form:

Show details

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Rod Moment CalculationThe moment of inertia calculation for auniform rod involves expressing an masselement in terms of a distance element dr

along the rod. To perform the integral, it isnecessar to express eve thing in the integralin terms of one variable, in this case thelength variable r. !ince the total length " hasmass #, then #$" is the proportion of mass tolength and the mass element can beexpressed as shown. Integrating from %"$& to

"$& from the center includes the entire rod.The integral is of pol nomial t pe(

Rod Moment About End)nce the moment of inertia of an ob*ect aboutits center of mass has been determined, themoment about an other axis can be determinedb the +arallel axis theorem (

In this case that becomes

This can be confirmed b direct integration

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Rotational-Linear 1arallels

Common Moments o' ,nertia

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Moment o' ,nertia 0$amplesMoment of inertia is defined with respect to a specific rotation a is. )he moment of inertia of a

point mass with respect to an a is is defined as the product of the mass times the distance from the

a is s*uared. )he moment of inertia of any e tended ob ect is built up from that basic definition.

)he general form of the moment of inertia involves an integral .

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Moments of inertia for commonforms

Where moment of inertia appears in physicalquantities

Moment o' ,nertia 4eneral ormSince the moment of inertia of an ordinary ob ect involves a continuous distribution of mass at a

continually varying distance from any rotation a is, the calculation of moments of inertia generally

involves calculus, the discipline of mathematics which can handle such continuous variables. Since

the moment of inertia of a point mass is defined by

then the moment of inertia contribution by an infinitesmal mass element dm has the same form. )his

+ind of mass element is called a differential element of mass and its moment of inertia is given by

0ote that the differential element of moment of inertia d( must always be defined with respect to a

specific rotation a is. )he sum over all these mass elements is called an integral over the mass.

Asually, the mass element dm will be e pressed in terms of the geometry of the ob ect, so that the

integration can be carried out over the ob ect as a whole for e ample, over a long uniform rod;.

Baving called this a general form, it is probably appropriate to point out that it is a general form onlyfor a es which may be called


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1arallel a$is theorem

http://en.)i2ipedia.org/)i2i/1arallel5a$is5theorem From @i+ipedia, the free encyclopedia

(n physics , the parallel a$is theorem can be used to determine the moment of inertia of a rigid ob ect about any a is, given the moment of inertia of the ob ect about the parallel a is through the

ob ect1scentre of mass and the perpendicular distance between the a es.

2et:

I CM denote the moment of inertia of the ob ect about the centre of mass,

M the ob ect1s mass and d the perpendicular distance between the two a es.

)hen the moment of inertia about the new a is z is given by:

)his rule can be applied with the stretch rule and perpendicular a es rule to find moments of inertia

for a variety of shapes.

%arallel a es rule for area moment of inertia.

)he parallel a es rule also applies to the second moment of area area moment of inertia;7

where:

I z is the area moment of inertia through the parallel a is,

I x is the area moment of inertia through the centre of mass of the area ,

A is the surface of the area, and

d is the distance from the new a is z to the centre of gravity of the area.

)he parallel a is theorem is one of several theorems referred to as +teiner*s theorem , after Da+ob

Steiner .

,n classical mechanics(n classical mechanics, the %arallel a is theorem can be generali8ed to calculate a new inertia tensor

6 i7 from an inertia tensor about a center of mass , i7 when the pivot point is a displacement a from the

center of mass:

J ij 5 I ij E M a ' ij G a ia j;

@here a is the displacement vector from the center of mass to the new a is.

@e can see that, for diagonal elements when i=j ;, displacements perpendicular to the a is of

rotation results in the above simplified version of the parallel a is theorem.

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ngular Momentum o' a 1article)he angular momentum of a particle of mass m withrespect to a chosen origin is given by

2 5 mvr sin Hor more formally by the vector product

2 5 r p)he direction is given by the right hand rule which

would give 2 the direction out of the diagram. For an

orbit, angular momentum is conserved , and this leads

to one of Iepler1s laws . For a circular orbit, 2 becomes

2 5 mvr

ngular Momentum)he angular momentum of a rigid ob ect is defined as the product of the moment of inertia and the

angular velocity . (t is analogous to linear momentum and is sub ect to the fundamental constraints of

the conservation of angular momentum principle if there is no e ternal tor*ue on the ob ect. ngular

momentum is a vector *uantity . (t is derivable from the e pression for the angular momentum of a

particle

3omparison of linear and angular momentum

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ngular and Linear Momentumngular momentum and linear momentum are e amples of the parallels between linear and rotational

motion. )hey have the same form and are sub ect to the fundamental constraints of conservation

laws , the conservation of momentum and the conservation of angular momentum .

Rotational 8inetic 0nergy)he +inetic energy of a rotating ob ect is analogous to linear +inetic energy and can be e pressed in

terms of the moment of inertia and angular velocity . )he total +inetic energy of an e tended ob ect

can be e pressed as the sum of the translational +inetic energy of the center of mass and therotational +inetic energy about the center of mass. For a given fi ed a is of rotation, the rotational

+inetic energy can be e pressed in the form:

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)he e pressions for rotational and linear +inetic energy can be developed in a parallel manner from

the wor+-energy principle . 3onsider the following parallel between a constant tor*ue e erted on a

flywheel with moment of inertia ( and a constant force e erted on a mass m, both starting from rest.

For the linear case, starting from rest, the acceleration from 0ewton1s second law is e*ual to the

final velocity divided by the time and the average velocity is half the final velocity, showing that the

wor+ done on the bloc+ gives it a +inetic energy e*ual to the wor+ done. For the rotational case,

also starting from rest, the rotational wor+ is JH and the angular acceleration K given to the flywheel

is obtained from 0ewton1s second law for rotation . )he angular acceleration is e*ual to the final

angular velocity divided by the time and the average angular velocity is e*ual to half the final

angular velocity. (t follows that the rotational +inetic energy given to the flywheel is e*ual to the

wor+ done by the tor*ue.

9or2-0nergy 1rinciple

)he wor+-energy principle is a general principle which can be applied specifically to rotating

ob ects. For pure rotation, the net wor+ is e*ual to the change in rotational +inetic energy :

For a constant tor*ue , the wor+ can be e pressed as

and for a net tor*ue, 0ewton1s 'nd law for rotation gives

3ombining this last e pression with the wor+-energy principle gives a useful relationship for

describing rotational motion .

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Rolling b7ects (n describing the motion of rolling ob ects, it must be

+ept in mind that the +inetic energy is divided between

linear +inetic energy and rotational +inetic energy .

nother +ey is that for rolling without slipping, the

linear velocity of the center of mass is e*ual to the

angular velocity times the radius.

+phere on incline. ; &oop and cylinder on incline.

8inetic 0nergy o' Rolling b7ect(f an ob ect is rolling without slipping, then its +inetic energy can be e pressed as the sum of the

translational +inetic energy of its center of mass plus the rotational +inetic energy about the center

of mass. )he angular velocity is of course related to the linear velocity of the center of mass, so the

energy can be e pressed in terms of either of them as the problem dictates, such as in the rolling of

an ob ect down an incline. 0ote that the moment of inertia used must be the moment of inertia

about the center of mass. (f it is +nown about some other a is, then the parallel a is theorem may be

used to obtain the needed moment of inertia.

Rotation ector 0$ampleshttp://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html , )his is an active graphic. 3lic+ on any e ample.

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45871255 Rotational Motion

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