Chapter 11ww2.odu.edu/~agodunov/teaching/notes231/Chapter_11.pdf · Chapter 11 Equilibrium 2 Two...
Transcript of Chapter 11ww2.odu.edu/~agodunov/teaching/notes231/Chapter_11.pdf · Chapter 11 Equilibrium 2 Two...
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Chapter 11
Equilibrium
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Two Types of equilibrium
Static equilibriumA body is at rest (no translation or rotation)example: a book resting on a desk
DynamicA body is moving without accelerationexample: airplane flying with constant speed
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Stable static equilibriumIf a body returns to a state of static equilibrium after having been displaced from it by a forceexample: a marble at the bottom of a hemispherical bowl
Unstable static equilibriumIf a small force can displace the body and end the equilibriumexample: a marble at the top of a hemispherical surface
Static equilibrium: stable or unstable
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Two Conditions for Equilibrium
1. The vector sum of all the external forces that act on the body must be zero
2. The vector sum of all the external torques that act on the body, measured about any possible point, must be zero
0=netFr
0=netτr
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the conditions in the component form
balance of forces balance of torques
000000
==
==
==
net,znet,z
net,ynet,y
net,xnet,x
τFτFτF
in physics 231 we consider only situations in which the forces that act on the body lie in the (x,y) plane, then
⎪⎩
⎪⎨
⎧
=
=
=
000
net,z
net,y
net,x
τFF
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Choice of a reference point for torque
find torque about point O if torque about point P is given
It does not matter what point to choose, but choose wisely
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The center of gravityThe gravitational force on an extended body is the vector sum of the gravitational forces acting on the individual elements (atoms) of the body. Instead of counting all those individual elements, we can say
The gravitational force acting on a body effectively acts at a single point, called the center of gravity of the body
“effectively” means that if the forces on individual elements were turned off and force at the center of gravity were turned on the net force and the net torque (about any point) acting on the body would not change
gFr
gFr
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Finding the center of gravityIf is the same for all elements of a body, then the body’s center of gravity is coincident with the body’s center of mass.
Proof:
gr
∑∑
∑∑∑
∑∑
==
×=
=×=×=×=
×=
===
iiicm
ii
cmnet
iii
iii
iiinet
iii
ii
iig
rmM
rmM
Mr
gMgrmM
grmgmr
gmr
gMmggmF
rr
rr
rrrrrrrr
rrr
rrrr
1
1
and where
torque net the
element single a on torque
:force naltranslatio net the
τ
τ
τ
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A uniform beam on two scalesA uniform beam, of length L and mass m is at rest with it’s ends on two scales. A uniform block, of mass M, is at rest on the beam, with it’s center a distance x from the beam’s left end. What do the scales read?
example
left scale right scale
Mm
xL
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A uniform beam on two scalesexample
x L/2
gMrgmr
lFr
rFr
02/00
=+−−
=−−+
rl
rl
LFLmgxMgFmgMgFF
:point left the about : yalong
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A uniform beam on two scalesexample
⎟⎠⎞
⎜⎝⎛ +=⎟
⎠⎞
⎜⎝⎛ +=
=
⎟⎠⎞
⎜⎝⎛ +=⎟
⎠⎞
⎜⎝⎛=
=
⎟⎠⎞
⎜⎝⎛ −
+=
⎟⎠⎞
⎜⎝⎛ +=
MmgFMmgF
Lx
MmgFmgF
x
MLxLmgF
MLxmgF
lr
lr
l
r
21
21
21
21
2/21
21
02121
and
:2 case
and
:1 case
forces unknown two for system the solving
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Lifting a carYou are about to lift a car of mass M using a lever. If you mass is m and the lever is L meters long and negligible mass comparing to m and M . Where must the support point be placed?How much load must the support hold?
example
gMmNmM
mLx
xMgmgxLMgmgN
)(
0)(0
+=+
=
=−−
=−−
and then
:O :y
lift a planet?
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in a museum …
Two identical pictures are hanging from a ceiling. Where is the greatest tension in the string?
question
1T 2T 3T 4T
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A picture on a wall
A picture can be hung on a wall in three different ways, as shown. Where is the greatest tension in the string?
example
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Not too contemporary art
Toy penguins are hanging from a ceiling. Each crossbar is horizontal, has negligible mass, and extends three times as far to the right of the wire supporting it as to the left. Penguin 1 has mass 48 kg. What are the masses of the other penguins?
example
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A problem with a safeA safe of mass M hanging by a rope from a boom with dimensions a and b. The boom consist of a hinged beam and a horizontal cable that connects the beam to a wall. The uniform beam has a mass m. The masses of the cable and the rope are negligible.
a) what is the tension in the cable
b) find the magnitude of the net force on the beam from the hinge.
example
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Equilibrium: comments on problem solvingChoose ONE object in a time for consideration
Draw a free-body diagram (show ALL forces acting ON that object)
Choose (wisely) a coordinate system and resolve forces in their components
“Generate” equilibrium equations using the conditions for equilibrium
⎪⎩
⎪⎨
⎧
=
=
=
000
net,z
net,y
net,x
τFF
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A problem with a safeexample
⎟⎠⎞
⎜⎝⎛ +=
=−−
+==
mMabgT
bmgbMgaT
MgmgFTF
c
c
v
ch
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021
equation third the from
:O
:y :x
beam :object
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An advertising signA metal sign of mass M is suspended from the end of a rod of mass m and length L. The rod is supported by a cable with negligible mass.
a) what is the tension in the cable
b) find the magnitude of the net force on the rod from the hinge.
example
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An advertising signexample
gMrgmr
Tr
xFryF
r
αh
L
02
sin
0sin0cos
=−−
=−−+=−
LMgmgLLT
MgmgTFTF
y
x
α
ααrod :object
⎟⎠⎞
⎜⎝⎛ += MmgT
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sinα
solution
22sin
Lhh+
=α where
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what if …example
gMrgmr
Tr
xFryF
r
αh
L
( ) 02
sin
0sin0cos
=−−−
=−−+=−
LMgmgLTxL
MgmgTFTF
y
x
α
αα
⎟⎠⎞
⎜⎝⎛ +
−= Mm
xLLgT
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sinα
solution
( )22sin
xLh
h
−+=α where
x
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An advertising sign (more complicated)A metal sign of mass M is suspended by two wires from a rod of mass m and length L. The rod is supported by a cable with negligible mass.
What is the tension in the cable
example
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An advertising sign (more complicated)example
gM r
2
gmr
TryF
r
xFr
gM r
2
( ) 02
)(2
sin
022
sin
0cos
=+−−−
=−−−+
=−
gMLaLmgLLT
gMgMmgTF
TF
y
x
α
α
αrod :object
( )
for
solution
⎟⎠⎞
⎜⎝⎛ +==
−+=
MmgTa
aLMmLLgT
2sin0
)2(sin2
α
α
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A pickup truckA pickup truck has a wheelbase of L meters. Ordinary M1 kg rests on the front wheels, and M2 on the rear wheels, when the truck is parked on a level road. A box of m kg is now placed on the tailgate, x meters behind the rear axel.
How much total weight now rests on the front wheels?
On the rear wheels?
How much weight would need to be placed on the tailgate to make the front wheels come off the ground?
example
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example
gM r1
gM r2
gmr
1Nr
2Nr
L x
0)(
00
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2121
=−=
=−−
=−−−+
mLxMgN
xmgLNgLMmggMgMNN
for solution
axle) second the (aroung
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Climbing ladderexample