Name (no name, no credit!) : Bradley j. nartowt Meeting time (circle one): all of them!
Quiz #9 – gravitation
Problem: Point masses masses are distributed distributed as seen below. Mass A has a mass of m, mass B has a mass of 2m, and mass C has a mass of 8m. A fourth mass D (not pictured) pictured) of mass mass 10m is placed placed at (x,y) coordinates (-3d, -4d). Newton’s gravitational constant is G. Question 1 (2 point): Compute the gravitational potential energy of mass D in terms of m, d, G, and necessary constants.
U U AD U BD U CD
U
Gm A mD r AD
GmB mD rBD
GmC mD r CD
Gm2
1 10 2 10 8 10 d ( 3) 3) 2 ( 4) 4) 2 ( 3) 3) 2 ( 5) 5) 2 ( 1. 1.5) 2 ( 4) 4) 2
Gm2
Gm2 2 d 24.2 d 34 18.25 20
80
Question 2 (1.5 points): If masses B, B, C, and D are fixed fixed in place and mass A is released released from from rest, what is the x-component x-component of acceleration acceleration experienced experienced by A? Answer Answer in terms of m, d, G, G, and necessary necessary constants. constants.
a x
F x
m A
F x, AC F x , AD mA
G
m A mC r AC
2
m m rˆ AC xˆ G r A 2D rˆAD xˆ AD
mA
m ( xˆ cos yˆ sin ) xˆ m D xˆ cos yˆ sin xˆ G C 2 2 2 (1.5d ) 2 [ (3d ) ( 4d ) ] a x
Gm 8(1 0 0)
d 2
(1.5)
2
10 cos tan 25
1 4 d 3 d
Gm 8
4 2 costan 1 d 2.25 5 3 2
B, C, and D are fixed fixed in place and mass A is released released from from rest, what is the Question 3 (1.5 points): If masses B, y-component y-component of acceleration acceleration experienced experienced by A? Answer Answer in terms of m, d, G, G, and necessary necessary constants. constants. a y
F
y
m A
Fy , AB F y , AD mA
G
m A mB r AB
2
m m rˆ AB yˆ G r A 2D rˆAD yˆ AD
mA tan 1 43 d d
10m Gm 2 2m ˆ sin 2 2(0 0 1 1) sin G 2 ( x cos 2 yˆ sin 2 ) yˆ 25d 2 5 d d
Gm
2 4 2 sin tan 1 ; d 5 3 2
Bonus (+0.50 points): Compute the gravitational potential energy of the whole system of masses A, B, C, and D in the original Problem. Hint: there are 3 + 2 + 1 = 6 terms to consider, which is the solution to the problem “if four (4) people all shook hands with each other, and nobody shakes each other’s hand twice, what is the total number of handshakes?”. Half handshakes?”. Half of this problem was already done in Question 1:
U U question1 U AB U AC U BC U question 1 Ubonus ; So the question really becomes “compute U bonus ”, which is pretty much a repeat of question-1. BOR-ING! U bonus
Gm A mB r AB Gm2
Gm A mC rAC
GmB mC rBC
Gm2
1 2 1 8 28 d 02 12 (1.5)2 02 (1.5)2 12
Gm2 2 1.5 d 16.2 d 3.25 8
16
Adding the above, and the answer from part 1, U bonus
Gm2 d
24.2
Gm2 d
16.2
Gm2 d
40.4