Chapter 5 - SteadyServerPages
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Chapter 5
β’ Applying
Newtonβs Laws
Equilibrium
An object is in equilibrium when
the net force acting on it is zero.
In component form, this is
The net force on each
man in the tower is
zero.
Slide 5-13
Equilibrium
A hanging street sign with more than one force acting on it
πΉ1
πΉ2
πΉ3
πΉπ₯1 + πΉπ₯2 + πΉπ₯3 =0
πΉπ¦1 + πΉπ¦2 + πΉπ¦3 =0
Equilibrium
What are the components of the forces?
πΉ1
πΉπ¦1
πΉπ₯1
πΉπ¦3
πΉπ₯2
πΉ2
0
πΉπ₯1 + πΉπ₯2 + πΉπ₯3 = 0
πΉ3
0
πΉπ¦1 + πΉπ¦2 + πΉπ¦3 =0
Slide 5-14
Example Problem
A 100 kg block with a weight of 980 N hangs on a rope. Find
the tension in the rope if
A. the block is stationary.
B. itβs moving upward at a steady speed of 5 m/s.
C. itβs accelerating upward at 5 m/s2.
Slide 5-15
Example Problem
A 100 kg block with a weight of 980 N hangs on a rope. Find
the tension in the rope if
A.
the block is stationary.
πΉπ¦ = 0
πΉπ¦ = ππ
+ πΉπ = 0
ππ
100kg
ππ
+ ππ = 0
m
ππ
= 100kg β 9.8 2 = 980N
s
πΉπ
Slide 5-15
A 100 kg block with a weight of 980 N hangs on a rope. Find the tension in the
rope if
A.
B.
C.
the block is stationary.
itβs moving upward at a steady speed of 5 m/s.
itβs accelerating upward at 5 m/s2.
πΉπ¦ = 0
πΉπ¦ = ππ
+ πΉπ = 0
ππ
π£
100kg
ππ
+ ππ = 0
m
ππ
= 100kg β 9.8 2 = 980N
s
πΉπ
Slide 5-15
A 100 kg block with a weight of 980 N hangs on a rope. Find the tension in the
rope if
A.
B.
C.
the block is stationary.
itβs moving upward at a steady speed of 5 m/s.
itβs accelerating upward at 5 m/s2.
πΉπ¦ = π ππ¦
ππ
π
ππ
+ πΉπ = πππ¦
100kg
ππ
= πππ¦ β ππ
ππ
= π ππ¦ β π
m
m
ππ
= 100kg 5 2 β β9.8 2
s
s
= 1480N
πΉπ
Slide 5-15
Example Problem
A wooden box, with a mass of 22 kg, is pulled at a constant
speed with a rope that makes an angle of 25° with the wooden
floor. If the coefficient of kinetic friction is ππ = .5, what is the
tension in the rope?
Slide 5-16
A wooden box, with a mass of 22 kg, is pulled at a constant
speed with a rope that makes an angle of 25° with the wooden
floor. If the coefficient of kinetic friction is ππ = .5, what is the
tension in the rope?
πΉπ¦ = ππ
π¦ + πΉπ + π = 0
πΉπ₯ = ππ
π₯ + πΉπ = 0
ππ
π¦ = βππ β π
ππ
π₯ = βππ π
ππ
π
πΉπ
25°
πΉπ
ππ
π¦
ππ
π₯
Slide 5-16
A wooden box, with a mass of 22 kg, is pulled at a constant speed with a rope that makes
an angle of 25° with the wooden floor. If the coefficient of kinetic friction is ππ = .5, what is
the tension in the rope?
ππ
π₯ = βππ π
ππ
π¦ = βππ β π
ππ
cos π = βππ π
ππ
sin π = βππ β π
ππ
cos π
ππ
sin π = βππ +
ππ
ππ
cos π
= βπ
ππ
ππ
π
πΉπ
25°
πΉπ
ππ
π¦
ππ
π₯
Slide 5-16
A wooden box, with a mass of 22 kg, is pulled at a constant speed with a rope that makes
an angle of 25° with the wooden floor. If the coefficient of kinetic friction is ππ = .5, what is
the tension in the rope?
ππ
cos π
ππ
sin π = βππ +
ππ
ππ
cos π
ππ
sin π β
= βππ
ππ
ππ
cos π
sin π β
= βππ
ππ
βππ
ππ
=
cos π
sin π β
ππ
Example Problem
A ball weighing 50 N is pulled back by a rope to an angle of 20°.
What is the tension in the pulling rope?
Slide 5-19
Example Problem
A ball weighing 50 N is pulled back by a rope to an angle of 20°.
What is the tension in the pulling rope?
ππ
ππ
π¦
π
πΉπ
ππ
π₯
Slide 5-19
A ball weighing 50 N is pulled back by a rope to an angle of 20°.
What is the tension in the pulling rope?
πΉπ₯ = ππ
π₯ + π = 0
ππ
π₯ = βπ
π
π
π¦
°
cos 20 =
ππ
π₯
πΉπ¦ = ππ
π¦ + πΉπ = 0
ππ
π¦ = βππ = 50N
ππ
π¦
ππ
π
πΉπ
ππ
π₯
Slide 5-19
Using Newtonβs Second Law
Slide 5-20
Dynamics with the 2nd law
π
ππ¦
πΉs
ππ₯
Ξ£πΉπ₯ = πΉsπ₯ + π€π₯ = πππ₯
π€
Ξ£πΉπ¦ = πΉsπ¦ + π€π¦ = πππ¦
Example Problem
A sled with a mass of 20 kg slides along frictionless ice at 4.5 m/s.
It then crosses a rough patch of snow which exerts a friction force
of -12 N. How far does it slide on the snow before coming to rest?
Slide 5-21
A sled with a mass of 20 kg slides along frictionless ice at 4.5 m/s.
It then crosses a rough patch of snow which exerts a friction force
of -12 N. How far does it slide on the snow before coming to rest?
π£π₯
Slide 5-21
A sled with a mass of 20 kg slides along frictionless ice at 4.5 m/s.
It then crosses a rough patch of snow which exerts a friction force
of -12 N. How far does it slide on the snow before coming to rest?
πΉπ
Slide 5-21
A sled with a mass of 20 kg slides along frictionless ice at 4.5 m/s.
It then crosses a rough patch of snow which exerts a friction force
of -12 N. How far does it slide on the snow before coming to rest?
2
ππ
=
2
βππ
2
ππ
πΉ = ππ
+ 2πβπ₯
πΉ
=2
βπ₯
π
βππ2 π
βπ₯ =
2πΉ
βπ₯
πΉ
π=
π
Slide 5-21
Mass and Weight
β’ Mass and weight are not the same
π€ = ππ
m
The moonβs gravity
β’ The moon has about 1/6 of the
gravity of earth
π€m =
1
π 6π
m
Weightlessness
β’ Falling doesnβt mean you have no
weight
β’ But this is the term we use anyway
π£
π
Mass and Weight
βw = may = m(βg)
w = mg
Slide 5-23
Which of the following statements
about mass and weight is correct?
.
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25% 25% 25% 25%
ur
e
A. Your mass is a measure of the
force gravity exerts on you
B. Your mass is same everywhere
in the universe
C. Your weight is the same
everywhere in the universe
D. Your weight is a measure of
your resistance of being
accelerated
30
Apparent Weight
Slide 5-24
Example Problem
A 50 kg student gets in a 1000 kg elevator at rest. As the elevator
begins to move, she has an apparent weight of 600 N for the first 3
s. How far has the elevator moved, and in which direction, at the
end of 3 s?
Slide 5-25
A 50 kg student gets in an elevator at rest. As the elevator begins to
move, she has an apparent weight of 600 N for the first 3 s. How far
has the elevator moved, and in which direction, at the end of 3 s?
πΉπ¦ = π€π + π€π = ππ
π€π = 600N
π€π
π
π€π + π€π
π=
π
π€π = ππ
1 2
1 2
βπ¦ = π£ππ¦ π‘ + ππ‘ β βπ¦ = ππ‘
2
2
0
1 π€π + π€π
βπ¦ =
2
π
π‘2
π€π
Slide 5-25
Normal Forces
πΉN
m
π€
πΉN = ππ
Normal Forces
+π¦
πΉN
π
+π₯
β π€ cos π
π€
π
β π€ sin π
Normal Forces
π€ = ππ
πΉN
+π¦
+π₯
Ξ£πΉπ₯ = βππ sin π = πππ₯
Ξ£πΉπ¦ = βππ cos π + πΉN = πππ¦
βππ cos π + πΉN = 0
πΉN = ππ cos π
π
β π€ cos π
π€
π
β π€ sin π
Static Friction
fs max = µsn
Slide 5-30
Kinetic Friction
fk = µkn
Slide 5-31
Coefficients of static friction are typically greater
than coefficients of kinetic friction.
50%
50%
Fa
lse
Tr
u
e
A. True
B. False
Include static friction
πΉf = ππ πΉN
πΉN
π
+π¦
+π₯
β π€ cos π
π€
π
β π€ sin π
Kinetic and Rolling friction
πΉf = ππ πΉN
πΉf = ππ πΉN
π£
π£
πΉf
πΉf
Working with Friction Forces
Slide 5-32
Drag
An object moving in a gas or liquid experiences a drag force
Drag coefficient. Depends on
details of the objectβs shape.
βStreamliningβ reduces drag by
making CD smaller. For a typical
object, CD 0.5.
Density of gas or liquid. Air has
a density of 1.29 kg/m3.
A is the objectβs cross section area
when facing into the wind.
Drag depends on the square of the speed.
This is a really important factor that limits the
top speed of cars and bicycles. Going twice
as fast requires 4 times as much force and,
as weβll see later, 8 times as much power.
Slide 5-34
Drag
π£
πΉπ·
πΉπ· =
1
πΆπ ππ΄π£ 2
2
Direction opposite of motion
Drag
πΉπ· =
1
ππ΄π£ 2
4
π£
π is the fluid density
A is the cross-sectional area
πΆπ is the drag coefficient(most objects
have πΆπ = 1/2)
Terminal Speed
Sky divers go no faster
than this
Itβs about 150 mph for
average objects
At this speedβ¦
Ξ£πΉπ¦ = 0
πΉπ·
π£π‘
Example Problem
A car traveling at 20 m/s stops in a distance of 50 m. Assume that
the deceleration is constant. The coefficients of friction between a
passenger and the seat are ΞΌs = 0.5 and ΞΌk = 0.3. Could a 70 kg
passenger slide off the seat if not wearing a seat belt?
Slide 5-33
Example Problem
A car traveling at 20 m/s stops in a distance of 50 m. Assume that
the deceleration is constant. The coefficients of friction between a
passenger and the seat are ΞΌs = 0.5 and ΞΌk = 0.3. Could a 70 kg
passenger slide off the seat if not wearing a seat belt?
m
π£ = 20
s
βπ₯ = 50m
Slide 5-33
Example Problem
A car traveling at 20 m/s stops in a distance of 50 m. Assume that
the deceleration is constant. The coefficients of friction between a
passenger and the seat are ΞΌs = 0.5 and ΞΌk = 0.3. Could a 70 kg
passenger slide off the seat if not wearing a seat belt?
m
π£ = 20
s
βπ₯ = 50m
0
ππ2 = ππ2 + 2πβπ₯
π=
2
βππ
2βπ₯
πΉπ = ππ π = ππ ππ
ππ β· ππ ππ
Slide 5-33
Cross-Section Area
Slide 5-35
Terminal Speed
A falling object speeds up
until reaching terminal speed,
then falls at that speed
without further change.
If two objects have the same
size and shape, the more
massive object has a larger
terminal speed.
At terminal speed, the net
force is zero and the object
falls at constant speed with
zero acceleration.
Slide 5-36
Summer 2014 Beginning of lecture
Applying Newtonβs Third Law: Interacting Objects
Slide 5-37
Example Problem
Block A has a mass of 1 kg; block Bβs mass is 4 kg. They are
pushed with a force of magnitude 10 N across a frictionless
surface.
a. What is the acceleration of the blocks?
b. With what force does A push on B? B push on A?
Slide 5-39
Block A has a mass of 1 kg; block Bβs mass is 4 kg. They are
pushed with a force of magnitude 10 N across a frictionless
surface.
a. What is the acceleration of the blocks?
b. With what force does A push on B? B push on A?
πΉπ₯ = 10N = 1kg + 4kg ππ₯
10N
m
ππ₯ =
=2 2
5kg
s
10N
Slide 5-39
Block A has a mass of 1 kg; block Bβs mass is 4 kg. They are
pushed with a force of magnitude 10 N across a frictionless
surface.
a. What is the acceleration of the blocks?
b. With what force does B push on A? A push on B?
Force from block A
on hand
Force from hand on
block A
Slide 5-39
Block A has a mass of 1 kg; block Bβs mass is 4 kg. They are
pushed with a force of magnitude 10 N across a frictionless
surface.
a. What is the acceleration of the blocks?
b. With what force does A push on B? B push on A?
πΉπ₯π΄ = πΉβπ΄ + πΉπ΅π΄ = 1kg
m
2 2
s
10N + πΉπ΅π΄ = 2N
Force from hand on
block A
πΉπ΅π΄ = β8N
Force from block B
on block A
Slide 5-39
Block A has a mass of 1 kg; block Bβs mass is 4 kg. They are pushed with
a force of magnitude 10 N across a frictionless surface.
a. What is the acceleration of the blocks?
b. With what force does A push on B? B push on A?
πΉπ₯π΅ = πΉπ΄π΅ = ππ΅ π
From Newtonβs 3rd law
πΉπ΄π΅ = βπΉπ΅π΄
βπΉπ΅π΄ = ππ΅ π
β β8N = ππ΅ π
8N = ππ΅ π = 4kg
m
2 2
s
Force from block B
on block A
Force from block A
on block B
Slide 5-39
Which pair of forces is a βthird law pairβ?
The string tension and the friction force acting on A
The normal force on A due to B and the weight of A
The normal force on A due to B and the weight of B
The friction force acting on A and the friction force acting on B
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25% 25% 25% 25%
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A.
B.
C.
D.
Example Problem
What is the acceleration of block B?
Slide 5-42
Sum the forces on block B?
πΉπ₯π΅ = 20N + πΉπ΄π΅ = ππ΅ ππ΅
πΉπ΄π΅ = πΉπ = ππ ππ΅π΄ = ππ ππ΄ π
20N + ππ ππ΄ π
m
ππ΅ =
= 5.764 2
ππ΅
s
Slide 5-42
Ropes and Pulleys
All ropes are assumed massless with
uniform tension
All pulleys are assumed frictionless
π
π
Example Problem
Block A, with mass 4.0 kg, sits on a frictionless table. Block B, with
mass 2.0 kg, hangs from a rope connected through a pulley to
block A. What is the acceleration of block A?
Slide 5-44
Block A, with mass 4.0 kg, sits on a frictionless table. Block B, with
mass 2.0 kg, hangs from a rope connected through a pulley to
block A. What is the acceleration of block A?
π
Same string, same tension??
ππ΄
πΉππ΄
πΉπ = ππ π = 0 β π
ππ΅
πΉππ΅
Slide 5-44
Block A, with mass 4.0 kg, sits on a frictionless table. Block B, with mass 2.0 kg,
hangs from a rope connected through a pulley to block A. What is the acceleration
of block A? SUM THE FORCES on each mass separately
π₯
πΉπ₯π΄ = ππ΄ = ππ΄ ππ΄
πΉπ¦π΅ = ππ΅ + πΉππ΅ = ππ΅ ππ΅
Note that the motions of these two masses are coupled
π¦
ππ΄
ππ΅
πΉππ΅
π₯
ππ΄
πΉππ΄
πΉππ΄
ππ΅
πΉππ΅
π¦
π₯
Slide 5-44
Block A, with mass 4.0 kg, sits on a frictionless table. Block B, with
mass 2.0 kg, hangs from a rope connected through a pulley to
block A. What is the acceleration of block A?
ππ΄ = ππ΄ ππ΄
ππ΅ + πΉππ΅ = ππ΅ ππ΅
βππ΄ ππ΄ + πΉππ΅ = ππ΅ ππ΅
ππ΄ = βππ΅
Newtonβs Third
Butβ¦ ππ΄ = ππ΅
π¦
π₯
So just call them both π
βππ΄ π + πΉππ΅ = ππ΅ π
Now solve for π
π
πΉππ΄
π
πΉππ΅ π₯
π¦
πΉππ΅
2.0kg β π 1
π=
=
= π
ππ΄ + ππ
6.0kg
3
Slide 5-44
Summary
Slide 5-45
Summary
Slide 5-46