CONSERVATION OF MOMENTUM Name INTWO DIMENSIONS 1. Two objects are arranged on a level, frictionless table as shown. Two experiments are conducted in which object A is launched toward the stationary block B. The initial speed of object A is the same in both experiments; the direction is not. The initid and final velocities of object A in each experiment are shown. The mass of block B is four times that of object A (m" = 4m^). Top views Velocity Yectors (drawn to scale) Mech HW-63 { do, Experiment 1 - before collision ld",l= o ilur: ? - after collision Tn, . uei Direction of Lfio Experiment2 - before collision 2 - after collision a. In the space provided, draw separate arrows represeiting the direction of the change in momentum vector of object A in the two experiments. Is the magnitude of the change in m.omcntum of object A in experiment I greater than, less than, or equal to that in experiment 2? Explain. Experiment I Experiment 2 b. In the space provided, draw separate arrows representing the direction of the change in momentum vector of block B in the two experiments. Afier the collisions, is the magnitude of the momentum of block B in experiment I greater than, less than, or equal to that in experiment 2? If the momentum of block B is zero in either case, state that explicitly. Explain. Direction of Al^ Experiment I Experiment 2 Tutorials in Introductory Physics McDermott, Shaffer, and the P,E.G., U. Wash. I arson Custom zod EA.,for U,CO, Boulder Mech HW-64 Conseroation of momentum in two dimensions Two objects collide on a level, frictionless table. The mass of object A is 5.0 kg; the mass of object n is f .O tg. The objects stick together after the collision. The initial velocity of object A and the final velocity of both objects are shown. 2. Before collision dltAi After collision 1do, = d"r) (One side of a square represents 0.1 m/s) In the space provided, draw separate arrows for object A and for object B representing the direction of the change in momentum vector of the object. Is the magnitude of the change in m,omentum of object A greater than, less than, or equal to that of object B? Explain your reasoning. Direction of Al Object A Object B b. System C is the system of both objects A and B combined. How does the momentum of system C before the collision compare to the momentum of system C after the collision? Discuss both magnitude and direction. Construct and label a vector showing the momentum 6f system C at an instant before the collision. Show your work clearly. c, Construct and label a vector showing the initial velocity of object B. Show your work clearly. (Each side of a square represents 0.4 kg'm/s) (Each side of a square represents 0.1 m/s) Tutorials in Introductory Physics McDermott, Shaffer, and the P.E.G., U, Wash. @ rson Custom znd H., for U,CO, Boulder Consentation of momentum in two dimensions Name 3. Object A collides on a horizontal frictionless surface with an Frictionles.

1. CONSERVATION OF MOMENTUM Name
INTWO DIMENSIONS
1. Two objects are arranged on a level, frictionless table as
shown. Two experiments are conducted
in which object A is launched toward the stationary block B.
The initial speed of object A is the
same in both experiments; the direction is not. The initid and
final velocities of object A in each
experiment are shown.
The mass of block B is four times that of object A (m" = 4m^).
Top views Velocity Yectors
(drawn to scale)
Mech
HW-63
{
do,
Experiment 1 - before collision
ld",l= o ilur: ?
- after collision
Tn,
. uei
Direction of Lfio

2. Experiment2 - before collision 2 - after collision
a. In the space provided, draw separate arrows
represeiting the direction of the change in momentum
vector of object A in the two experiments.
Is the magnitude of the change in m.omcntum of
object A in experiment I greater than, less than, or
equal to that in experiment 2? Explain.
Experiment I Experiment 2
b. In the space provided, draw separate arrows
representing the direction of the change in momentum
vector of block B in the two experiments.
Afier the collisions, is the magnitude of the momentum
of block B in experiment I greater than, less than, or
equal to that in experiment 2? If the momentum of
block B is zero in either case, state that explicitly.
Explain.
Direction of Al^
Experiment I Experiment 2
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Mech

3. HW-64
Conseroation of momentum in two dimensions
Two objects collide on a level, frictionless table. The mass of
object A is 5.0 kg; the mass of
object n is f .O tg. The objects stick together after the collision.
The initial velocity of object A
and the final velocity of both objects are shown.
2.
Before collision
dltAi
After collision
1do, = d"r)
(One side of a square represents 0.1 m/s)
In the space provided, draw separate arrows for
object A and for object B representing the direction of
the change in momentum vector of the object.
Is the magnitude of the change in m,omentum of
object A greater than, less than, or equal to that of
object B? Explain your reasoning.
Direction of Al
Object A Object B
b. System C is the system of both objects A and B combined.
How does
the momentum of system C before the collision compare to the
momentum of system C after the collision? Discuss both
magnitude

4. and direction.
Construct and label a vector showing the momentum 6f system
C at an
instant before the collision. Show your work clearly.
c, Construct and label a vector showing the initial velocity of
object B.
Show your work clearly.
(Each side of a square
represents 0.4 kg'm/s)
(Each side of a square
represents 0.1 m/s)
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Consentation of momentum in two dimensions Name
3. Object A collides on a horizontal frictionless surface with an
Frictionless horizontal surface
Mech
HW-65
initially stationary target, object X. The initial and final
velocities
of object A are shown. The final velocity of object X is not
given.

5. a, At an instant during the collision, is the net force on object A
zero ot nbn-zero?
b. During the collision, is the momentum of object A
conserved?
Explain.
Is the momentum of the system consisting of objects A and
X conserved? Explain.
c. On the same horizontal surface, object C collides with an
initially
stationary target, objectZ, The initial speeds of objects C and A
are
the same, ild trtx= trlz) tne.= tltc,
After the collisions, object C moves in the direction shown and
has
the same final speed as object A.
i. In the space below, copy the vectors d6; and d6l with their
tails
togbther. Use these vectors to draw the change in velocity
vector for glider C, AAc.
Top view
Before collision
A X
oH
ax at rest
Aftercollision

6. AX€o
6o,
Arter collision
ii. Is the magnitude of the change in velocity vector of object A
greater than, less than, or
equal to the magnitude of the change in velocity vector of
object C? Explain.
iii. Is the magnitude of the change in momentum vector of
object A greater than, less than,
or equal to the magnitude of the change in momentum vector of
object C? Explain.
iv. Is the final speed of object X greater than, less than, or equal
to the final speed of
objectZ? Explain.
d. Consider the following incorrect statement:
'6liders A ond C have the some chonge in momentum. They
hove the some moss, ond because
they have the some initiql speed ond same f inol speed, Av is
the some for eoch of ,them."
Discuss the error(s) in the reasoning.
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Before collision

7. Z
o
At rest
-
DYNAMICS OF RIGID BODIES Name Mech
HW-69
1. Energy analysis of the block-and-spool
problem
A block and a spool are each pulled
across a level, frictionless surface by a
string, as illustrated at right.
The string pulling the block is tied to a
small hook at the center of the front face
of the block (not shown). The string
pulling the spool is wrapped many times
around the spool and may unwind as it is
pulled.
The block and the spool have the same mass. The strings are
pulled with the szlme constant tension and start pulling at the
same instant.
Make the approximation that the strings and the hook are
massless.
a. Does the spool cross the finish line before, afier, or at the
sante instant as the block? Explain.

8. Tbp view
Start Finish
k-- -_-_- d -- *--4
Bloc
Spooll
u
Same
b. Consider the following dialogue between two students:
Student l: "f think thot there's the some omount of work done on
block ond spool os they
ore pulled from the stort to the f inish since they both move the
some distonce."
Student 2: 'f disogree. f think thot the hond pullirg the spool
does more work thon the
hond pulling the block since the string unwinds qs the spool is
pulled."
With which student, if either, do you agree? Explin.
When each crosses the finish line, is the total kinetic energy of
the spool greater than, Iess
than, or equal to that of the block? Explain. (Hint: Use the
work-energy theorem.)
d. When each crosses the finish line, is the translational kinetic
energy of the spool greater
than,less than, or equal to that of the block? Explain.
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Mech Dynamics of rigid.bodies
HW-70
2. Three identical rectangular blocks are at rest on a level,
frictionless surface. Forces of equal
magnitude that act in the same direction are exerted on each of
the three blocks. Each force is
exerted at a different point on the block (indicated by the
symbol "Xo'), as shown in the top-view
diagram below. The location of each block's center of mass is
indicated by a small circle.
For each of the blocks, draw an affow on the diagram above to
indicate the direction of the
acceleration of the block's center of mass at the instant shown.
If the magnitude of the
acceleration of the center of mass of any block is zero, state
that explicitly. Explain.
b. Rank the blocts according to magnitude of center-of-mass
acceleration, fromlargest to
smallest. If any two blocks have the same magnitude center-of-
mass acceleration, state so
explicitly. Support your ranking by drawinga point free-body
diagram for each block.
3, A uniform rigid rod rests on a level, frictionless surface. The
diagram below indicates four
different combinations of (1) net force on the rod and (2) net

10. torque on the rod about its center of
mass. In each box, draw vectors that represent one or two forces
that achieve the given
combination of net force and net torque. If any combination is
not possible, state so explicitly.
For example: In the second case, indicate one or two forces that
could be exerted on the rod so
that at the instant shown the net force on it is zero, but the net
torque on it is not zero.
Tbp view
li,,l = o, l(-,1 = o l{,,*l = o, l(,,1 * o l{,,,1 + o, li,,l = o lF".,l *
o, l?".,1 * o
.,,
Block I
Tbp view
Block 2 Block 3
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Dynamics of rigid bodies Name Mech
HW-71

11. 4. Three objects of equal mass, A, B, and C, are released from
rest
at the same instant from the same height on identical ramps.
Objects A and B are both blocks, and they slide down their
respective ramps without rotating. Object C rolls down the
ramp without slipping. Its moment of inertia is unknown.
Objects A, B, and C are made of different materials, thus the
coefficients of friction between the objects and their
coffesponding ramps are not necessarily the same.
Object A reaches the bottom of its ramp first, followed by
objects B and C, which reach the
bottom at the same instant.
a. Rank the objects according to magnitude of center-of-mass
acceleration,from largest to
smallest. If any objects have the same magnitude center-of-mass
acceleration, state so
explicitly. Explain.
b. Rank the net forces exerted on the three objects according to
magnitude, from largest to
smallest. If the net force on arly two objects is the same, state
so explicitly. Explain.
c. In the spaces
provided, draw and
label a (point) free-
body diagram for
each object.
Free-body diagraur
for object i
F-ree-bocly dia*uram

12. for object B
Free-body diagram
for object C
d. Rank the frictional forces exerted on the three objects
according to magnitude, from largest to
smallest. If the magnitude of the frictional force is the same on
any two objects, state so
explicitly. Explain your reasoning.
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All three objects
are relea.sed
Mech Dynamics of rigiilbodies
HW:72
5. Energy analysis of falling-spools experiment
The modified Atwood's machine shown at right consists
of two identical spools connected by a massless, ' "
inextensible thread that runs over an ideal pulley. The
thread is wrapped around spool A many times, but it is
attached to a fixed point on spool B, so that spool B will
not rotate.

13. The spools are released from rest from the same height
at the same instant.
a. In tutorial, you observed the motion of the spools
after they were released. Ignoring small dffirences
in their nntions:
. In which direction did each spool move?
. Did spool A hit the ground before, after, or at the same instant
as spool B?
Is the mngnitude of the center-of-nutss acceleration of spool A
(while it is fallin g) greater
than,less than, or equal to that of spool B? Explain.
Is the translational kinetic energy of spool A just before it hits
the ground greater thsn, less
than, or equal to that of spool B? Explain.
d. Is the total kinetic energy of spool A just before it hitsothe
ground greater than, less than, or
equal to that of spool B? Explain.
Consider the system consisting of all of these objects: spool A,
spool B, the thread, the
pulley, and the Eafth.
i. Explain how you can tell that the total energy of this
system(i.e., Ugnv,o+ Ug^",n * Kton,,e
* K,*nr. n * Kro' a, * K,o' s) is constant as spools A and B fall.
b.
c.

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Dynamics of rtgtd bodies Narne Mech
HW-73
Suppose that this system starts with Ugo", o = f/srr, B = 9 J.
Just before the spools hit the
ground, which is where the zero for gravitational potential
energy is chosen, spool A has
translational kinetic energy K,on,"a = 4 J. Determine the value
of the rotational kinetic
energy of spool A at this instant. Show your work.
6. A third identical spool, spool C, is added to
the falling-spools experiment described in the
preceding problem.
As above, all spools are released from rest
from the same height at the same instant.
Spool C is not in contact with any other
objects as it falls.
a, Rank the spools according to magnitude
of center -of-mas s acc eleration (while
falling), from largest to smallest. If any
spools have the same center-of-mass
acceleration, state so explicitly. Explain.
b. As in the preceding problem, suppose that Ugrrr. n = Ugr,, s
= 9 J before the spools are

15. released. Just before the spools hit the ground, which is where
the zero for gravitational
potential energy is chosen, spool A has trarrslational kinetic
energy K,*,, e,= 4J.
i. Rank the spools according to maximum tanslatianal kinetic
energy, from largest to
smallest. If any spools have the same maximrrtn translational
kinetic energy, state so
explicitly. Explain. (Use the definition K*, = lma"^'.)
ii. Rank the spools according to maximum total kinetic energy,
from largest to smallest. If
any spools have the same maximum total kinetic energy, state
so explicitly. Explain.
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CONSERVATION OF
ANGULARMOMENTUM
Name Mech
HW-77
1. In tutorial, you observed the following three experiments
involving a
student sitting at rest on a stool, holding a spinning bicycle
wheel as
shown at right:

16. Experiment 1: The student places his arm against the side of the
wheel, slowing it to half its initial angular speed.
Experiment 2: The student places his arm against the side of the
wheel, bringing it to a stop.
Experiment 3: The student quickly flips the wheel over (so that
it is
spinning clockwise when viewed from above, with
the same angular speed it had initially).
Student initially at rest
Initial sense
of wheel's
rotation
a. You observed that the final angular speed of the student in
experiment 3 is greater than that
in experiment2, Account for this result using the ideas
developed in the tutorial.
b. Rank the experiments according to final kinetic energy of the
wheel, from largest to smallest.
If the final kinetic energy of the wheel is the same in any two
experiments, state so explicitly.
(Hint: Can kinetic energy ever be negative?) Explain.
Rank the experiments according to final kinetic energy of the
student, from largest to
smallest. If the final kinetic energy of the student is the same in
any two experiments, state
soexplicitly. Explain.
d. Rank the followingfour quantities from largest to smallest:

17. the initial kinetic energy of the
wheel (K*i) and the final kinetic energy of the student-wheel
system in experiments 1 ,2, and
3 (Kstr, etc.). Explain. (Hint: It may be helpful to think about
changes in energy other than
mechanical energy.)
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-
Mech
HW-78
Conseruation of angular momentum
Z. The diagram below illustrates four hypothetical collisions
that take placg on a level, frictionless
surface.- The collisions are shown from a top-view perspective.
All pucks are identical. If a
linear or angular velocity is not specified, it is zero. If distances
appear to be equal, assume that
they are.
For each hypothetical collision:
a. Specify the direction of the angular momentum of the rod-
puck(s) system with respect to the
center of the rod both before and after the collision. If
necessary, use the convention that a

18. vector into the page is represented by the symbol I and a vector
out of the page is
represented by the symbol O.
b, Specify the direction of the linear momentum of the rod-
puck(s) system, both before and after
the collision.
c. On the basis of your answers above, state whether each
hypothetical collision could ot could
not occrfi. If a particular hypothetical collision could not occur,
state whether it violates
(l) the principle of conservation of linear momentum, (2) the
principle of conservation of
angular momentum, or (3) both.
Before collision
Case I
Case 2
Case 3
Case 4
Ball
sticks to
rod
Sense of
rotation
After collision

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You plan to create a new business, which you want to be
designated as minority and women owned (MWBD).Before you
can incorporate your company, you must research the following:
· How you can become certified as MWBD
· The standards you must uphold
· The percentage of women and/or minorities that you must
employ
· The benefits of operating such a business