10/14/2013phy 113 c fall 2013 -- lecture 141 phy 113 c general physics i 11 am-12:15 pm mwf olin 101...
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PHY 113 C Fall 2013 -- Lecture 14 110/14/2013
PHY 113 C General Physics I11 AM-12:15 PM MWF Olin 101
Plan for Lecture 14:
Chapter 12 – Static equilibrium
1. Balancing forces and torques; stability
2. Center of gravity
3. Will discuss elasticity in Lecture 15 (Chapter 15)
PHY 113 C Fall 2013 -- Lecture 14 210/14/2013
PHY 113 C Fall 2013 -- Lecture 14 310/14/2013
Newton’s law of gravitation:
Earth’s gravity:
Stable circular orbits of gravitational attracted objects:
RES
F
a
vMsat
Summary of gravity:
RE m
2226
2411
2
2
m/s8.9m/s)1037.6(
1098.51067.6
E
E
E
E
R
GMg
mgR
mGMF
212
122112
ˆ
r
mGm rF
ESsat
ES
satE
R
vM
R
MGMF
2
2
PHY 113 C Fall 2013 -- Lecture 14 410/14/2013
From Webassign Assignment #12:
When a falling meteoroid is at a distance above the Earth's surface of 3.40 times the Earth's radius, what is its acceleration due to the Earth's gravitation?m/s2 towards earth
r=4.4RE 22
2
4.4 E
EE
E
R
GM
r
GMa
mar
mGM
m
PHY 113 C Fall 2013 -- Lecture 14 510/14/2013
From Webassign Assignment #12:
An artificial satellite circles the Earth in a circular orbit at a location where the acceleration due to gravity is 6.79 m/s2. Determine the orbital period of the satellite.
r
s
T
a
GM
a
rT
rTr
Tr
r
v
r
vmma
r
mGM
E
E
6675
79.6
1098.51067.62
22
2/2
4/1
3
2411
4/1
3
222
2
2
PHY 113 C Fall 2013 -- Lecture 14 610/14/2013
From Webassign Assignment #12:
How much work is done by the Moon's gravitational field as a 1090 kg meteor comes in from outer space and impacts on the Moon's surface?
M
M
R
M
R
M
R
mGM
r
mGMdr
r
mGMW
MM
2
RM
iclicker questionA. W>0B. W<0
PHY 113 C Fall 2013 -- Lecture 14 710/14/2013
From Webassign Assignment #12:
A space probe is fired as a projectile from the Earth's surface with an initial speed of 1.74 104 m/s. What will its speed be when it is very far from the Earth? Ignore atmospheric friction and the rotation of the Earth.
vi22
2
1
2
1f
E
Ei
ffii
mvR
mGMmv
UKUK
PHY 113 C Fall 2013 -- Lecture 14 810/14/2013
From Webassign Assignment #12:Plaskett's binary system consists of two stars that revolve in a circular orbit about a center of mass midway between them. This statement implies that the masses of the two stars are equal (see figure below). Assume the orbital speed of each star is v = 190 km/s and the orbital period of each is 10.7 days. Find the mass M of each star. (For comparison, the mass of our Sun is 1.99 1030 kg.)
G
TvM
vTR
T
Rv
MTv
R
GM
R
vM
2
4
:algebra someAfter 2
2
know o want t;, Know
2
3
2
22
PHY 113 C Fall 2013 -- Lecture 14 910/14/2013
From Webassign Assignment #12:Plaskett's binary system consists of two stars that revolve in a circular orbit about a center of mass midway between them. This statement implies that the masses of the two stars are equal (see figure below). Assume the orbital speed of each star is v = 190 km/s and the orbital period of each is 10.7 days. Find the mass M of each star. (For comparison, the mass of our Sun is 1.99 1030 kg.)
iclicker exercise:Who might pose a question like this?
A. A mean professor.B. A puzzle master.C. An observational astronomer.
PHY 113 C Fall 2013 -- Lecture 14 1010/14/2013
Meanwhile – back on the surface of the Earth:
Conditions for stable equilibrium
0 : torqueof Balance
0 :force of Balance
ii
ii
τ
F
PHY 113 C Fall 2013 -- Lecture 14 1110/14/2013
Stability of “rigid bodies”
N
mig
PHY 113 C Fall 2013 -- Lecture 14 1210/14/2013
Center-of-mass
ii
iii
CM m
m rr
Torque on an extended object due to gravity (near surface of the earth) is the same as the torque on a point mass M located at the center of mass.
miri
jrjrτ Mggm CMi
ii
rCM
PHY 113 C Fall 2013 -- Lecture 14 1310/14/2013
Notion of stability:
mg(-j)
r
Tq
F=ma T- mg cos q 0
mg sin q maq
t=I a r mg sin q = mr2 a mraq
Notion of equilibrium: i
i 0F i
i 0τ
Example of stable equilibrium for q 0.
PHY 113 C Fall 2013 -- Lecture 14 1410/14/2013
Unstable equilibrium:
mg(-j)
r
Tq
Support above CM:
Support below CM:
PHY 113 C Fall 2013 -- Lecture 14 1510/14/2013
Nik Wallenda walking on high wire across Grand Canyon
PHY 113 C Fall 2013 -- Lecture 14 1610/14/2013
Analysis of stability: i
i 0F i
i 0τ
PHY 113 C Fall 2013 -- Lecture 14 1710/14/2013
0)1( :Torques
0 :Forces
gxmmgM
gmgmgMn
cD
PcD
**X
PHY 113 C Fall 2013 -- Lecture 14 1810/14/2013
PHY 113 C Fall 2013 -- Lecture 14 1910/14/2013
**X
Fg1
mg
RCM
0)()2( :Torques 1 CMg RmgmF
PHY 113 C Fall 2013 -- Lecture 14 2010/14/2013
iclicker question:
F1F2
Consider the above drawing of the two supports for a uniform plank which has a total weight Mg and has a weight mg at its end. What can you say about F1 and F2?
(a) F1 and F2 are both up as shown.
(b) F1 is up but F2 is down.
(c) F1 is down but F2 is up.
L/3
LMg mg
PHY 113 C Fall 2013 -- Lecture 14 2110/14/2013
F1F2
L/3
LMg mg
023
:Torques
0 :Forces
2
21
mgLL
MgL
F
mgMgFF
**X
mgMgFmgMgF 22
1 3
2
312
PHY 113 C Fall 2013 -- Lecture 14 2210/14/2013
iclicker question:The fact that we found F1<0 means:
A. We set up the problem incorrectlyB. The analysis is correct, but the
direction of F1 is opposite to the arrowC. Physics makes no sense
iclicker question:What would happen if we analyzed this problem by placing the pivot point at F1 ?:
A. The answer would be the same.B. The answer would be different.C. Physics makes no sense
PHY 113 C Fall 2013 -- Lecture 14 2310/14/2013
T
Mgmg
**X
2/
x
sin
2//
0sin2
0
MgmgxT
TMgmgx
NT
NMgNmg
mxmo
313
200 600
2 8 53For
PHY 113 C Fall 2013 -- Lecture 14 2410/14/2013
d
**X
hr
hrrmg
hr
mgdF
hrFmgd
22
02
0
22
PHY 113 C Fall 2013 -- Lecture 14 2510/14/2013
mg
Mg
Fwall
N
T
Mg = 120 N
mg = 98 N
T < 110 N
**X
q
sin
cos2/
0sin
cos2
cos
0
0
MgmgxT
LF
MgLmgxL
MgmgN
FT
wall
wall
PHY 113 C Fall 2013 -- Lecture 14 2610/14/2013
x
A ladder of weight Mg and of length L is supported by the ground with static friction force f and by a frictionless wall as shown. The firefighter has weight mg and is half-way up the ladder. Find the force that the ladder exerts on the wall.
q
wallF
f**X
h
amg
Lh
axMgfF
amgxL
aMghF
fF
wall
wall
wall
2
02
1
0