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Unformatted text preview: Physics 171.101 Exam 2 SOLUTIONS Nov, 17, 1999
Prof. Barnett 1. The adjacent ﬁgure _shows two students sitting on a massless,
5 meter long plank. Rashidat Agbaje, MR 2 60 kg, is on the
extreme right end of the plank, and Lucas Finn, M L = 60 kg, is on the extreme left end of the plank. The plank is suspended (*2 meters"!
from a wire 2 meters from the left end. Take I g[ = 10m / sec2 and
consider all students are to be “point masses” throughout
this problem. m-€ A) (9 pts) What is the torque about the point P where the wire connects to the plank?
’Fcotal = 'FR + ’F’L = 600 newton meters into page B) (9 pts) What is the angular acceleration, 62’, of the plank about point P?
"F : Io'Z —> 62 = 'F/I '
I = MLRi + M3122 2 60(4 + 9) = 780 kg m2
5i 2 0.77 radians/seconds2 into the page. C) (9 pts) What is the acceleration, ER, of Rashiclat?
|a| : [al RRaahidat : (3) X 0.77 = 2.3 "rm/sec2 El: direction is downward. D) (9 pts) What is the acceleration, ECM, of the center of mass of Rashidat and Lucas?
Their center of mass is 0.5 meters to the right of the wire. 5 : 0.5 X 0.77 = 0.385 m/sec2
downward. E) (9 pts) What is the tension of the wire?
Ftotal I mtotal X 50M 2 W + T = T ~ mg T = m([g|— lat):120 kg X (10 — 0.385) m/se62 : 1154. newtons plank
2. As shown in the adjacent ﬁgure, a six meter long plank (Ede
is attached vertically to a frictionless axle through its 5 m/sec
center. The total mass of plank is 30 kg and its moment of X; inertia about the axle is 90 kg m2. A point—mass student,
Anna Widmer, MA, mass = 50 kg, runs past at a speed of 5 m/sec and grabs onto the bottom end of the plank. After
she has grabbed the plank she and the plank rotate about the
axle with an initial instantaneous angular velocity L230. Anna
rises into the air, holding onto the plank, and eventually stops at some height, h, above the
ground when the plank reaches some maximum angle, 9mm. Anna A) (7 pts) Is there some point, P, in the “plank plus Anna” system about which there is no torque at the instant when Anna grabs the plank? If “Yes”. explain where the point, P,
is and why there is no torque.
There is no torque about the axle. The force of gravity on the plank acts at the center
of mass for the plank which is at the axle. The force of the axle in holding up the plank,
even if it has a horizontal component as required in part G below, is also at the axle. So
neither of these forces produce torques because the R is 0. The force of gravity on Anna
points down and is located directly below the axle, so her I? is parallel to her Two
parallel vectors have a cross product of 0. B) (8 pts) What are l) Anna’s momentum and 2) angular momentum about the point P at
the instant before she grads the plank? P 2 mV 2 50 5 : 250 kgm/sec to the right.
Lfg‘jﬁg’ = R X P : 750kgm2/sec out of the page. C) (8 pts) After Anna grabs the plank, What is the total moment of inertia of Anna. and the
plank about the point P?
[T = plank + IA
[A = MARE1 = 50 32 = 450kg m2
IT 2 90 + 450 = 540169 m2 D) (8 pts) What is the angular velocity, (30, of Anna and the plank about the axle immedi-
ately after she grabs the plank? Because there is no torque about the axle the angular momentum about it is conserved. a Therefore, L = E31551 found in part B. L = In a a = 5/1 = 750/540 = 1.39 radi—
ans / sec into the page. E) (8 pts) What is the total mechanical energy of Anna plus the plank at the instant after
she grabs the plank? KE' : ngz = 0.5 540 1.392 = 521 joules F) (8 pts) Find the maximum height, h, that Anna reaches above the ground due to the
rotation of the plank about the axle to its maximum angle, 49mm.
Use energy conservation with the realization that because the plank’s center of mass is at the axle it does not change position as Anna swings up. The only potential energy
change is due to Anna’s position’s change. KE' : mgh. —> h = 521/500 meters : 1.04 meters. G) (8 pts) Did the momentum of the “Anna plus plank” system change when Anna grabbed
the plank? If “Yes”, what impulse was exerted on the system?// Yes. it changed. The
momentum after the collision is Anna’s momentum plus the plank’s momentum. The
center of mass of the plank does not move so the total linear momentum of the plank is 0. Thus, the total linear momentum after the collision is just : mm,me =
5031.39 2 208.5 kg m/sec to the right. Impulse : I 2 A13 : Pix-ml — Pinmaz 2 208.5 — 250 : —41.5 kg m/sec. The impulse:
vector points to the left. This impulse comes from the axle pushing to the left on the plank when Anna grabs the plank. This force produces no torque about the axle, I? = 0,
so angular momentum is still conserved about the axle, as we said in part A. ...
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