free-body exercises: linear motion...free-body solutions: linear motion ... one-body and two-body...

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Free-Body Exercises: Linear Motion In each case the rock is acted on by one or more forces. All drawings are in a vertical plane, and friction is negligible except where noted. Draw accurate free-body diagrams showing all forces acting on the rock. LM-l is done as an example, using the "parallelogram" method ..For convenience, you may draw all forces acting at the center of mass, even though friction and normal reaction force act at the point of contact with the surface. Please use a ruler, and do it in pencil so you can correct mistakes. Label forces using the following symbols: w = weight of rock, T = tension, n = normal reaction force, f = friction. I __ ._. .__ ..__ .__~L _ LM-9. Rock is sliding on a frictionless incline.

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Page 1: Free-Body Exercises: Linear Motion...Free-Body Solutions: Linear Motion ... one-body and two-body problems in particle mechanics.Aprerequisitefor doing so is the analysis of all the

Free-Body Exercises: Linear MotionIn each case the rock is acted on by one or more forces. All drawings are in a vertical plane, and friction is negligible exceptwhere noted. Draw accurate free-body diagrams showing all forces acting on the rock. LM-l is done as an example, usingthe "parallelogram" method ..For convenience, you may draw all forces acting at the center of mass, even though friction andnormal reaction force act at the point of contact with the surface. Please use a ruler, and do it in pencil so you can correctmistakes. Label forces using the following symbols: w = weight of rock, T = tension, n = normal reaction force, f = friction.

I

__._..__. .__.__~L _

LM-9. Rock is sliding on a frictionlessincline.

Page 2: Free-Body Exercises: Linear Motion...Free-Body Solutions: Linear Motion ... one-body and two-body problems in particle mechanics.Aprerequisitefor doing so is the analysis of all the

LM-13. Rock is decelerating because ofkinetic friction.

LM-16. Rock is tied to a rope and pulledstraight upward, accelerating at 9.8 m1s2.

No friction.

LM-II. Rock is sliding at constant speedon a frictionless surface.

LM-14. Rock is rising in a parabolictrajectory.

/

II

LM-12. Rock is faIling at constant(terminal) velocity.

LM-15. Rock is at the top of a parabolictrajectory.

I

II

I

II

I

I

LM-18. Rock is tied to a rope and pulledso that it accelerates horizontally at 2g.No friction.

\ \ \

LM-17. Rock is tied to a rope and pulledso that it moves horizontally at constantvelocity. (There must be friction.)

Page 3: Free-Body Exercises: Linear Motion...Free-Body Solutions: Linear Motion ... one-body and two-body problems in particle mechanics.Aprerequisitefor doing so is the analysis of all the

Free-Body Solutions: Linear MotionThe dashed arrows and construction lines in these solutions are for explanation only, and not part of the finished diagram.The free-body diagram in each case consists of only the dark, solid arrows. Forces of the same magnitude or lines of thesame length are indicated by the same number of "tick" marks drawn through the two lines or arrows. Symbols: w = weight,T = tension, n = normal reaction force, f = friction.

T Note that the2 I tension is not

i proportional tostring length.

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::::=IIII

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f and n areactually I

! applied at '\!he surface)

I----~~~-----

Page 4: Free-Body Exercises: Linear Motion...Free-Body Solutions: Linear Motion ... one-body and two-body problems in particle mechanics.Aprerequisitefor doing so is the analysis of all the

/I

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T(= 2w)

I········HI··········IIIIII

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f- -. -~---,-_.-_.,-_.._.--..... '-_._.._-_.._-.--- .. --

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1-;.>-

,,

Resultant ofT and" is inI direction of

~leration)

Resultant of,T and w isequal tol

Page 5: Free-Body Exercises: Linear Motion...Free-Body Solutions: Linear Motion ... one-body and two-body problems in particle mechanics.Aprerequisitefor doing so is the analysis of all the

Free-Body Exercises: Circular MotionDraw free-body diagrams showing forces acting on the rock, and in each case, indicate the centripetal force. Please notethat the rock is not in equilibljum if it is moving in a circle. The centripetal force depends on angular velocity and theremay not be any indication of exactly how big that force should be drawn. Symbols: w = weight, T = tension, f = friction,n = normal reaction force, Fe = centripetal force.

CM-I. Swinging on a rope, at lowestposition. No friction.

CM-4. Rock is swinging on a rope. Nofriction.

CM-7. Rock is riding on a horizontal diskthat is rotating at constant speed about itsvertical axis. Friction prevents rock fromsliding. Rock is moving straight out of thepaper.

CM-2. Tied to a post and moving in acircle at constant speed on a fiictionlesshorizontal surface. Moving straight out ofthe paper.

CM-5. Rock is moving downward in avertical circle with the string horizontal.

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~\

CM-3. String is tied to a post. Rock ismoving toward you in a horizontal circle atconstant speed. No friction.

CM-6. Rock is swinging on a rope, at thetop of a vertical circle. No friction.

CM-9. Rock is stuckby frictionagainst theinside wall of a drum rotating about itsvertical axis at constant speed. Rock ismoving straight out of the paper.

CM-8. Rock is resting against the ifrictionless inside wall of a cone. It moves Iwith the cone, which rotates about its Ivertical axis at constant angular speed.

I

IiII

I

Page 6: Free-Body Exercises: Linear Motion...Free-Body Solutions: Linear Motion ... one-body and two-body problems in particle mechanics.Aprerequisitefor doing so is the analysis of all the

Free-Body Solutions: Circular Motion

.·· ··•·· .. · .. · .. ··•···.. ··.. ··.. ··.. ·· 1I

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//~c=T+WJ T

Page 7: Free-Body Exercises: Linear Motion...Free-Body Solutions: Linear Motion ... one-body and two-body problems in particle mechanics.Aprerequisitefor doing so is the analysis of all the

Exercises in Drawing and Utilizing Free-Body DiagramsKurt Fisher, Division of Natural Sciences and Mathematics, Dowling College, Oakdale, NY 11769-1999;[email protected]

Students taking the algebra-based introductory physics

course often have great difficultysetting up Newton's second lawequations of motion for dealing withone-body and two-body problems inparticle mechanics. A prerequisite fordoing so is the analysis of all the rel-evant forces, both the visible ones(those indicated or identified asapplied forces) and the unseen onessuch as gravity and friction. In turn,the tool for this force analysis is thefree-body diagram (FBD).

It would seem that if FBD's wereintroduced, and their application tothe generation of the requiredequations of motion illustrated, theclass would readily catch on. Unfor-tunately, it doesn't work that way.The analysis of forces is dependenton their being correctly perceived.The excellent free-body scale-draw-ing exercises by J. E. Court! are use-ful for developing the concepts ofvector resolution and force analysis.They can be used to firm up a stu-dent's understanding of FBD con-struction to accurate scale, therebygiving insight as to how weight, nor-mal force, and tension force vectorsrelate to one another. They are alsohelpful in diagnosing the persistenceof the naive "motion implies force"concepts that students have so muchtrouble shedding.

But FBD's alone are not enough!There must be the follow-through ofutilizing them. The short cut of grop-ing for formulas and numbers to pluginto them is seemingly too irre-sistible. Nonetheless, most problemscannot be solved correctly withoutdue analysis and, in any case, an ana-lytical attitude should be fostered asone of the main goals of education.

When required to include an FBDwith a problem, many students drawa perfunctory diagram that looks likea fully loaded pincushion. Vectors aremis-oriented, unlabeled, and/or showno directions. Such an FBD obvious-ly cannot be used to proceed to theequation of motion. Yet, even the stu-dents who get the FBD correct do notuse it further and will write an equa-tion of motion that obviously doesnot follow from their FBD, therebydefeating its very purpose. I can onlyascribe this resistance to analysis asthe manifestation of a learning stylethat stems from the "show-and-tell"methodology and the avoidance ofword problems and other integrativeactivities.

The small set of exercises offeredhere (Fig. 1) shows how I try to habit-uate the student to the analysis stepsneeded to successfully work outproblems in particle dynamics. Theidea is to present a series of gradedexercises in identifying forces, havethe student install them on an FBD,and then take the next step-writedown the ~F equations followingfrom the analysis. Inherent in theseexercises is the redundancy necessaryfor the learner to internalize theprocess. I have been using these exer-cises since returning from the 1993Rensselaer conference.2 I bundlethem together with the aforemen-tioned Court exercises and distributethem soon after starting the mechan-ics chapter. My students are given athree-week window in which to workboth sets of exercises and submitthem for homework credit. One revi-sion is permitted. Parts of these exer-cises appear on tests, so the value ofworking them out is appreciated by

, the students.

To introduce my students to thismulti-step procedure, three or four ofthe exercises are worked out in classusing Socratic dialogue as much aspossible. I emphasize that writingexpressions for ~Fx and ~Fy is anindispensable prerequisite to solvingone- and two-body mechanics prob-lems. This is where these exercisesgo a step beyond those that solelyinvolve drawing FBD's. For eachcase the ~F x expression is carried outas far as possible. This means that, inthe cases where friction is taken intoaccount, it is necessary to substituteinto the ~Fx expression the normalforce yielded by the ~Fy = 0 equation(we always assume that there is nomotion in the y-direction).

A very useful aid to sorting out therelevant forces in a mechanics prob-lem is the ONIBY table, which I haverecently begun to require as part ofthe solution on tests. The idea for thistable came from perusing one of theSocratic Dialogue Inducing (SDI)labs,3 another fallout gem traceableto the 1993 Rensselaer conference. Itboth mirrors and reinforces the FBD;an additional benefit is that it speedsup grading because all the force vec-tor values, magnitudes, and direc-tions are organized in one place. Itrequires the student to name the bodyeach force acts ON and the body BYwhich that force is exerted.

I can offer only anecdotal percep-tions to indicate that a larger fractionof my class takes the solutions tostandard mechanics problems fartherthan before. For more exercise sam-ples, cqntact me by mail or e-mail. Iwould appreciate any feedback as toresults stemming from the deploy-ment of these exercises, and wouldalso welcome any additions or modi-

Page 8: Free-Body Exercises: Linear Motion...Free-Body Solutions: Linear Motion ... one-body and two-body problems in particle mechanics.Aprerequisitefor doing so is the analysis of all the

ReferencesI. James E. Court, Phys. Teach.,

31, 104 (1993).

2. Conference on the IntroductoryPhysics Course (May 1993).The proceedings of this land-mark conference have beenpublished by John Wiley &Sons, under the auspices ofRensselaer Polytechnic Insti-

tute and the National ScienceFoundation (edited by JackWilson, ISBN O-nl-15557-8).

3. Richard R. Hake. private com-munication; See also Phys.Teach. 30,546 (1992).

fications found to improve theireffectiveness.

Graded Exercises in Drawing and Utilizing Free-Body DiagramsUsing a ruler, draw free-body diagrams (FBD's) showing all forces acting on each body. Coordinate directions are indicatedin the leading diagram of a sequence. Forces that are replaced by their x- and y- components should be shown canceled out.Then using each FBD as a guide, write down the '2.Fx and '2.Fy expressions, carrying them to the point where numerical val-ues might be substituted for Fa' m, e, ep, and JL.

I.ONE·BODY F-B Diagram (Show only x- and y-com- IFx= IFy=CONFIGURATIONS ponents of all forces acting ON the body)

1. Frictionless t.xlevel surface. ___0.____2. Level surface Fa

with friction. ~-Applied force at ... ----an angle e. 1..1

'I

~

3. Incline with friction.Applied force parallelto incline. Fa > WI!

II. TWO·BODY Take the x-axis to lie along the IFx = IF =CONFIGURATIONS direction of motion of each body.

y

(Assume ideal pulleys) See #4 for the exam Ie.

4. m1 is on a~

m1: m1:frictionless hor-izontal surface and isconnected to hanging m2 m2:mass m2 by a mass-less string.

5. Same as #4 except ml: m,:that m I experiencesfriction. ml: m2:

6. m I experiences J!41 m,: ml:friction.m1 sin e> m2 m2: m2:

Exercises in Drawing and Utilizing Free-Body Diagrams Vol. 37, Oct. 1999 THE PHYSICS TEACHER 435

Page 9: Free-Body Exercises: Linear Motion...Free-Body Solutions: Linear Motion ... one-body and two-body problems in particle mechanics.Aprerequisitefor doing so is the analysis of all the

Free-Body Diagrams Revisited - IIJames E. Court, City College of San Francisco, San Francisco, CA 94112

[Editor's Note: We reproduce here a continuation of the collection of free-body exercises sent us by Jim Court, the first part of whichwas published in October. At the request of Jim's widow, a colleague and friend of the author; Paul Hewitt (One San Antonio Place -

2D, San Francisco, CA 94113-4032). is acting as Jim's representative in this important contribution to the teaching of physics.]

Free-Body Exercises: Rotational EquilibriumAll of the beams and the packages have the same weight w, and they are "uniform," which means the weight can be appliedat the center. These systems are in equilibrium, so the net torque, the net vertical force and the net horizontal force are allzero. Symbols: w = weight, T = tension, n = normal reaction force at surface, V = vertical reaction force at hinge,H = horizontal reaction force at hinge,j = friction. RE-I is done as an example.

Page 10: Free-Body Exercises: Linear Motion...Free-Body Solutions: Linear Motion ... one-body and two-body problems in particle mechanics.Aprerequisitefor doing so is the analysis of all the

RE-13.Equilibrium

I-III

RE-12.Equilibrium

Page 11: Free-Body Exercises: Linear Motion...Free-Body Solutions: Linear Motion ... one-body and two-body problems in particle mechanics.Aprerequisitefor doing so is the analysis of all the

Free-Body Solutions: Rotational EquilibriumSymbols: w = weight. T = tension, n = normal reaction force at surface, V = vertical reaction force at hinge, H = horizontal reac-tion force at hinge, f = friction.

Page 12: Free-Body Exercises: Linear Motion...Free-Body Solutions: Linear Motion ... one-body and two-body problems in particle mechanics.Aprerequisitefor doing so is the analysis of all the

.......... /f .!)~

T··················flf·············1 ,

~I

Page 13: Free-Body Exercises: Linear Motion...Free-Body Solutions: Linear Motion ... one-body and two-body problems in particle mechanics.Aprerequisitefor doing so is the analysis of all the

Free-Body Exercises: Rotational Non-equilibriumIn each case, draw arrows representing all forces acting on the cylinder or the beam. The solid, uniform cylinders, the pack-ages suspended from them, and the uniform beams all have the same weight w. In all but one of these examples the objectis not in rotational equilibrium, i.e. the torques do not add up to zero. Symbols: T = tension, wand m = weight and mass ofcylinders, beams and packages, n = normal reaction force at surface, V = vertical force at hinge or axle, H = horizontal forceat hinge, a = acceleration. RN-I is done as an example.

RN-I. Cylinder is supported on africtionless horizontal axle.

r<= w-ma)

@~RN-4. Cylinder is rolling down a rough(not frictionless) incline.

RN-7. Beam is swinging down throughhorizontal position.

RN-5. Cylinder was released with zeroangular velocity on a frictionless incline. Isit rolling?

RN-3. String is tied to ceiling and wrappedaround cylinder. Cylinder is falling.

RN-6. Beam is slipping. Both wall andfloor are frictionless.

RN-9. Beam is falling on a smooth(frictionless) floor. If the beam is releasedfrom rest, what path does the c of m take?

Page 14: Free-Body Exercises: Linear Motion...Free-Body Solutions: Linear Motion ... one-body and two-body problems in particle mechanics.Aprerequisitefor doing so is the analysis of all the

Free-Body Solutions: Rotational Non-equilibriumSymbols: T = tension, wand m = weight and mass of cylinders, beams and packages, n = normal reaction force at surface,V = vertical force at hinge or axle, H = horizontal force at hinge, a = acceleration.

w and " both paslthrough the c of m,10 there il no torque.The cylinder IUdesdownhiU.

V must be <wbecause the c of mis acceleratingdownward.