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Unformatted text preview: Final Exam Physics 20 Fall, 2008 .50 1— UT/oxv 1
Note: This final is certainly long, and it has some problems that
should interest even the most adept: of you! Please do not panic or
worry about the difficulties. Instead, work your way along carefully,
but without dawdling too much. Be sure to show your work clearly.
Remember, the answer itself is not the point; how you analyze the situation is what matters. Just because you can’t solve a problem when you first look at it
means very little. Think about it; try to find one thing you are sure
you can calculate correctly, even if it does not obviously lead to the
answer. Do that, and then think some more. Very often things will
then become much more clear. To do well, you should work hard for the entire exam, resting a bit
now and again as needed. Believe it or not, every one of you can
score quite a few points by setting up even the hardest problems, so be sure not to neglect them. If you can’t figure out how to attack a certain problem, read it
carefully once more, and then go on to the next problem. Often you
will see how to attack it later when'yofftl'look back. Please be sure to give the units for your results. Be sure your exam contains 12 problems! Good Luck, and have a great holiday. l. (5 points). Find the angle between the vectors a = 3i — j + 2k, and b = i + 2j — k, and state whether
it is less than, equal to, or greater than 90 degrees.  : 3‘2"2=“}2 [W1 i Q are.ch @acgg‘tz/édle 7 3300 2. (5 points). A car begins from rest and accelerates in a straight line with a constant acceleration of
4m/32. How fast is the car traveling after it has covered 400 In? 4L“ ;1'622 31:»: 2) f: biz/4 dlL" ﬁne/m.
4f?“ (7‘71 9‘55 M/a, WOM 3. (5 points). A mass M2, rests on a horizontal frictionless surface. There is a second mass M1 resting on M2, as shown below. The coefﬁcient of static frictionbetween the two masses is us. Find the maximum force that can be exerted on the upper mass before it will begin to slip relative to the lower
mass. we 9 1 a v
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“Vi boll/.2, syrrwi F={'M.+/M& Maa flu/wJZ/x/m/J 4T “VD/Wm /c_Mr6 M24: M: F
I ‘ Ml’le; ﬂng8(MlT/4J L MJMJ [I +13} (M 4. (5 points). Two masses, m1 and m are connected by a massless cord, running over a massless, 23
frictionless pulley, as shown below. The masses are free to move. Find the tension in the cord. 7“ r: 71ml arm. 24‘7"
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M23 or: “(ﬂip/Mg "7; ZMl/Vlldl 5. (5 points). A mass m, moves in a circle on a horizontal frictionless table, with the circle centered on
a small hole in the table. Mass m is attached by a massless, frictionless cord to a second mass M,
hanging below the table, with the cord running through the hole in the table, as shown below. Given
that mass m completes each circle in a time of P seconds, ﬁnd the radius of the circle in which it must
move in order for M to remain at rest. SECTION II. These problems are a bit tougher, but should be well Within your
ability, if you keep at them. 6. (10 points). A block slides down a stationary inclined plane with an angle ('3). It is observed to move
with constant velocity while sliding down. The block is stopped and then projected up the incline with
initial speed V0. How far will it move up the plane before coming to rest? V  \S‘étﬂf’uc’h DobIA}
 A H .
/ V?— (curr Aﬂ‘zo (10 points). A small mass In, on a string of length leroves in a vertical circle. At the top of the
circle, the tension in the string is equal to half the weight of the mass. With What speed is the mass moving at the top of the circle? 8. (10 points). A long rope of mass per unit length 3 kg/m is moving through a horizontal frictionless
tube at a speed of 10 m/s. The tube has a 90degree bend to the left as shown. The tube is held in
place, and does not move. Find the net force exerted on the rope by the tube. —"' To g S'TMzﬂUWE
t Fateu v 7" 1*
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Frictionless Tube p“: 30933 A
AP 1 i303 ./O/;.j/£L ~300 VAT 9. (10 points). An object of mass 4m, is moving to, theright at speed V on a frictionless track, and is
struck head on by a second object of mass m, traveling to the left with speed 2 V. The collision is
perfectly elastic, and the objects continue to move on the track. Find the velocity of each of the masses in the laboratory frame, after the collision. Verify that momentum is conserved in the lab. v " " 2v
t” ——9» <—————— #L m / \/
Wat—.2 31'],va P? U (A (L113 A/‘FTf/LL/MD l, 6146/“: SECTION III. These are the toughest ones, BUT, if you keep your head and go
for partial credit, you can score a lot of points in this section. You should do problem # 10, and then choose either problem #11 or # 12 to work
on. Even if you work on both 10 and 11, you must indicate which one you want to have graded for credit. 10. (15 points). Masses M1 and M2 slide on horizontal tables with the same coefﬁcient of kinetic friction u between the masses and the tables. They are connected to a third mass M3 by means of
massless cords and frictionless pulleys, as shown below. Find the tension in the upper cord.
(HINT: Once you have four equations with 4 unknowns, d0n‘t bother with the algebra until you have checked every thing you have done on the entire exam, and having nothing better to do!) ’4‘ Ibrérn (2)
m3©«QT:M3 43: (3)
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4} —42 C) 4}: ﬂ{ 4—3571, we» (91 (3) v7 {MI+M2)}_ ,2/uhm,M.,3 lZMWL 3 5° {MW/“IL % 2Mng
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53/41 M3 alT 2 QMIM13+QMiMzﬂj2 {Ir—M’IJ3'21V’IML 1“ 3W» ‘7” L7m1+ FF] 2/1”; M.) Remember to choose 1 of these last 2 problems to work on. Even {fyou work on both of them, you
must indicate which one you want to have graded. 11. (20 points). Two equal mass particles are free to slide on a frictionless track, which rotates about
an axis perpendicular to the plane of the page. The track rotates with uniform angular velocity 0), and
you may ignore gravity. The two masses are connectedby a massless rod of length L, as shown below. Hint: The expression for acceleration in polar coordinates is given by: a = m — ram 606+ 21:9). (a) Find the differential equation obeyed by r(t), the distance between the innermost mass and the axis
of rotation. (b.) Show that a solution to this equation is given by: 1’0“) = A6“ + Be‘m  L / 2 , provided
that, 06 has the correct value. What must this value be? (0.) Find the two constants A , and B for the case where the particles are released from rest (radial
velocity 3 0) at t = 0, with the inner particle on the axis, and ﬁnd r(t) for this case, i.e. plug your results
for A and B into your result for r(t). /, M
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,7.
v 7" 507%, ) (Wt/1M4”
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{ ‘FHZZL/“mﬂﬁf w/ T [J Jm/ fit/Lt/ Lg ca Z
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ﬂ/ﬂl Aéﬂﬂ‘ 82% :1: I ? TX “La 3—}: J/mi (Fang/“ﬂ My wwﬁggm/tg  12. (20 points). Calculate the volume and the center of mass position for a solid cone, of half angle 0'.
and height H. Use the coordinate system shown in the ﬁgure, to determine the x, y , and z coordinates
of the center of mass. Express your result for the volume in terms of H, and the radius of the cone, R. ...
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 Spring '08
 Freedman
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