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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.
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LM-9. Rock is sliding on a frictionless incline.
LM-13. Rock is decelerating because of kinetic friction.
LM-16. Rock is tied to a rope and pulled straight upward, accelerating at 9.8 m1s^2_._ No friction.
LM-II. Rock is sliding at constant speed on a frictionless surface.
LM-14. Rock is rising in a parabolic trajectory.
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LM-12. Rock is faIling at constant (terminal) velocity.
LM-15. Rock is at the top of a parabolic trajectory.
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LM-18. Rock is tied to a rope and pulled so that it accelerates horizontally at 2g. No friction.
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LM-17. Rock is tied to a rope and pulled so that it moves horizontally at constant velocity. (There must be friction.)
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T(= 2w)
I········HI·········· II II II ::!:: II II
f- -. -~---,-.-.,-...--..... '-...-.._-.--- .. -- ,I 1-;.>- ,,
Resultant of T and" is in I direction of ~leration)
Resultant of, T and w is equal tol
Free-Body Exercises: Circular Motion
Draw free-body diagrams showing forces acting on the rock, and in each case, indicate the centripetal force. Please note that the rock is not in equilibljum if it is moving in a circle. The centripetal force depends on angular velocity and there may 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 lowest position. No friction.
CM-4. Rock is swinging on a rope. No friction.
CM-7. Rock is riding on a horizontal disk that is rotating at constant speed about its vertical axis. Friction prevents rock from sliding. Rock is moving straight out of the paper.
CM-2. Tied to a post and moving in a circle at constant speed on a fiictionless horizontal surface. Moving straight out of the paper.
CM-5. Rock is moving downward in a vertical circle with the string horizontal.
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CM-3. String is tied to a post. Rock is moving toward you in a horizontal circle at constant speed. No friction.
CM-6. Rock is swinging on a rope, at the top of a vertical circle. No friction.
CM-9. Rock is stuck by frictionagainst the inside wall of a drum rotating about its vertical axis at constant speed. Rock is moving straight out of the paper.
CM-8. Rock is resting against the i frictionless inside wall of a cone. It moves I with the cone, which rotates about its I vertical axis at constant angular speed.
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Exercises in Drawing and Utilizing Free-Body Diagrams
Kurt Fisher, Division of Natural Sciences and Mathematics, Dowling College, Oakdale, NY 11769-1999;
fisherk@dowling.edu
S
tudents taking the algebra-
based introductory physics
course often have great difficulty
setting up Newton's second law
equations of motion for dealing with
one-body and two-body problems in
particle mechanics. A prerequisite for
doing so is the analysis of all the rel-
evant forces, both the visible ones
(those indicated or identified as
applied forces) and the unseen ones
such as gravity and friction. In turn,
the tool for this force analysis is the
free-body diagram (FBD).
It would seem that if FBD's were
introduced, and their application to
the generation of the required
equations of motion illustrated, the
class would readily catch on. Unfor-
tunately, it doesn't work that way.
The analysis of forces is dependent
on their being correctly perceived.
The excellent free-body scale-draw-
ing exercises by J. E. Court! are use-
ful for developing the concepts of
vector resolution and force analysis.
They can be used to firm up a stu-
dent's understanding of FBD con-
struction to accurate scale, thereby
giving insight as to how weight, nor-
mal force, and tension force vectors
relate to one another. They are also
helpful in diagnosing the persistence
of the naive "motion implies force"
concepts that students have so much
trouble shedding.
But FBD's alone are not enough!
There must be the follow-through of
utilizing them. The short cut of grop-
ing for formulas and numbers to plug
into them is seemingly too irre-
sistible. Nonetheless, most problems
cannot be solved correctly without
due analysis and, in any case, an ana-
lytical attitude should be fostered as
one of the main goals of education.
When required to include an FBD
with a problem, many students draw
a perfunctory diagram that looks like
a fully loaded pincushion. Vectors are
mis-oriented, unlabeled, and/or show
no directions. Such an FBD obvious-
ly cannot be used to proceed to the
equation of motion. Yet, even the stu-
dents who get the FBD correct do not
use it further and will write an equa-
tion of motion that obviously does
not follow from their FBD, thereby
defeating its very purpose. I can only
ascribe this resistance to analysis as
the manifestation of a learning style
that stems from the "show-and-tell"
methodology and the avoidance of
word problems and other integrative
activities.
The small set of exercises offered
here (Fig. 1) shows how I try to habit-
uate the student to the analysis steps
needed to successfully work out
problems in particle dynamics. The
idea is to present a series of graded
exercises in identifying forces, have
the student install them on an FBD,
and then take the next step-write
down the ~F equations following
from the analysis. Inherent in these
exercises is the redundancy necessary
for the learner to internalize the
process. I have been using these exer-
cises since returning from the 1993
Rensselaer conference.^2 I bundle
them together with the aforemen-
tioned Court exercises and distribute
them soon after starting the mechan-
ics chapter. My students are given a
three-week window in which to work
both sets of exercises and submit
them for homework credit. One revi-
sion is permitted. Parts of these exer-
cises appear on tests, so the value of
working them out is appreciated by
, the students.
To introduce my students to this
multi-step procedure, three or four of
the exercises are worked out in class
using Socratic dialogue as much as
possible. I emphasize that writing
expressions for ~Fx and ~Fy is an
indispensable prerequisite to solving
one- and two-body mechanics prob-
lems. This is where these exercises
go a step beyond those that solely
involve drawing FBD's. For each
case the ~F x expression is carried out
as far as possible. This means that, in
the cases where friction is taken into
account, it is necessary to substitute
into the ~Fx expression the normal
force yielded by the ~Fy = 0 equation
(we always assume that there is no
motion in the y-direction).
A very useful aid to sorting out the
relevant forces in a mechanics prob-
lem is the ONIBY table, which I have
recently begun to require as part of
the solution on tests. The idea for this
table came from perusing one of the
Socratic Dialogue Inducing (SDI)
labs,^3 another fallout gem traceable
to the 1993 Rensselaer conference. It
both mirrors and reinforces the FBD;
an additional benefit is that it speeds
up grading because all the force vec-
tor values, magnitudes, and direc-
tions are organized in one place. It
requires the student to name the body
each force acts ON and the body BY
which that force is exerted.
I can offer only anecdotal percep-
tions to indicate that a larger fraction
of my class takes the solutions to
standard mechanics problems farther
than before. For more exercise sam-
ples, cqntact me by mail or e-mail. I
would appreciate any feedback as to
results stemming from the deploy-
ment of these exercises, and would
also welcome any additions or modi-
References
I. James E. Court, Phys. Teach., 31, 104 (1993).
- Conference on the Introductory Physics Course (May 1993). The proceedings of this land- mark conference have been published by John Wiley & Sons, under the auspices of Rensselaer Polytechnic Insti-
tute and the National Science Foundation (edited by Jack Wilson, ISBN O-nl-15557-8).
- Richard R. Hake. private com- munication; See also Phys. Teach. 30,546 (1992).
fications found to improve their effectiveness.
Graded Exercises in Drawing and Utilizing Free-Body Diagrams Using a ruler, draw free-body diagrams (FBD's) showing all forces acting on each body. Coordinate directions are indicated in 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
level surface.^ t.x ___0.____
2. Level surface Fa
with friction. Applied force at (^) ... ~- ----
an angle e. 1..
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3. Incline with friction.
Applied force parallel
to 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. m 1 is on a
m 1 : m 1 : frictionless hor- izontal surface and is connected to hanging (^) m 2 m2: mass m 2 by a mass- less string.
- Same as #4 except (^) ml: m,: that m I experiences friction. (^) _ml: m 2 :
- m_ I experiences
J!
m,: ml: friction.
m 1 sin e> m 2 m2: m2:
Exercises in Drawing and Utilizing Free-Body Diagrams Vol. 37, Oct. 1999 THE PHYSICS TEACHER 435
RE-13. Equilibrium
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RE-12. II Equilibrium
Free-Body Solutions: Rotational Equilibrium
Symbols: 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.
Free-Body Exercises: Rotational Non-equilibrium
In 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 object is not in rotational equilibrium, i.e. the torques do not add up to zero. Symbols: 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. RN-I is done as an example.
RN-I. Cylinder is supported on a frictionless horizontal axle.
r<= w-ma)
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RN-4. Cylinder is rolling down a rough (not frictionless) incline.
RN-7. Beam is swinging down through horizontal position.
RN-5. Cylinder was released with zero angular velocity on a frictionless incline. Is it rolling?
RN-3. String is tied to ceiling and wrapped around cylinder. Cylinder is falling.
RN-6. Beam is slipping. Both wall and floor are frictionless.
RN-9. Beam is falling on a smooth (frictionless) floor. If the beam is released from rest, what path does the c of m take?
Free-Body Solutions: Rotational Non-equilibrium
Symbols: 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 pasl through the c of m, 10 there il no torque. The cylinder IUdes downhiU.
V must be <w because the c of m is accelerating downward.
