Transcript Lecture-08-09

Chapter 6
Applications of
Newton’s Laws
Midterm # 1
Average = 16.35
Standard Deviation = 3.10
Well done!
2
Who Wins?
Kevin Parks (5’6”, 90kg) takes a
handoff from Michael Rocco, but
the hole closes as he hits the line.
He is pinned between to O-linemen
behind him, and 3 D-linemen
ahead... and everyone is pushing,
lifting, grabbing, and kicking. His
acceleration is 1 m/s2 upfield.
What is the net force on KP?
a) I need to know a lot more about angles,
masses of the line men, etc.
b) I could answer this if only I had the
initial velocity
c) I would also need to know what planet
this happened on.
d) 90 N, upfield.
e) This isn’t the Tech game, is it?
3
Who Wins?
Kevin Parks (5’6”, 90kg) takes a
handoff from Michael Rocco, but
the hole closes as he hits the line.
He is pinned between to O-linemen
behind him, and 3 D-linemen
ahead... and everyone is pushing,
lifting, grabbing, and kicking. His
acceleration is 1 m/s2 upfield.
What is the net force on KP?
a) I need to know a lot more about angles,
masses of the line men, etc.
b) I could answer this if only I had the
initial velocity
c) I would also need to know what planet
this happened on.
d) 90 N, upfield.
e) This isn’t the Tech game, is it?
If you know the acceleration of a
body, you know the direction of the
net force. If you also know the
mass, then you know the
magnitude.
4
Will It Budge?
A box of weight 100 N is
at rest on a floor where
μs = 0.4. A rope is
attached to the box and
pulled horizontally with
tension T = 30 N. Which
way does the box move?
a) moves to the left, because the force of static
friction is larger than the applied force
b) moves to the right, because the applied force
is larger than the static friction force
c) the box does not move, because the static
friction force is larger than the applied force
d) the box does not move, because the static
friction force is exactly equal the applied force
e) The answer depends on the value for μk.
Static friction
(s = 0.4 )
m
T
5
Will It Budge?
A box of weight 100 N is
at rest on a floor where
μs = 0.4. A rope is
attached to the box and
pulled horizontally with
tension T = 30 N. Which
way does the box move?
a) moves to the left, because the force of static
friction is larger than the applied force
b) moves to the right, because the applied force
is larger than the static friction force
c) the box does not move, because the static
friction force is larger than the applied force
d) the box does not move, because the static
friction force is exactly equal the applied force
e) The answer depends on the value for μk.
The static friction force has a
maximum of sN = 40 N. The
tension in the rope is only 30 N. So
Static friction
(s = 0.4 )
m
T
the pulling force is not big enough
to overcome friction.
Follow-up: What happens if the tension is 35 N? What about 45 N?
6
Springs
Hooke’s law for springs states that the
force increases with the amount the spring
is stretched or compressed:
The constant k is called the spring constant.
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Springs
Note: we are discussing the force of the
spring on the mass. The force of the spring
on the wall are equal, and opposite.
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Springs and Tension
S1
S2
A mass M hangs on spring 1, stretching it length L1
Mass M hangs on spring 2, stretching it length L2
Now spring1 and spring2 are connected end-to-end,
and M1 is hung below. How far does the combined
spring stretch?
a) (L1 + L2) / 2
b) L1 or L2, whichever is smaller
c) L1 or L2, whichever is bigger
d) depends on which order the
springs are attached
e) L1 + L2
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Springs and Tension
A mass M hangs on spring 1, stretching it length L1
Mass M hangs on spring 2, stretching it length L2
Now spring1 and spring2 are connected end-to-end,
and M1 is hung below. How far does the combined
spring stretch?
S1
S2
Fs=T
W
a) (L1 + L2) / 2
b) L1 or L2, whichever is smaller
c) L1 or L2, whichever is bigger
d) depends on which order the
springs are attached
e) L1 + L2
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Springs and Tension
A mass M hangs on spring 1, stretching it length L1
Mass M hangs on spring 2, stretching it length L2
Now spring1 and spring2 are connected end-to-end,
and M1 is hung below. How far does the combined
spring stretch?
S1
S2
Fs=T
W
a) (L1 + L2) / 2
b) L1 or L2, whichever is smaller
c) L1 or L2, whichever is bigger
d) depends on which order the
springs are attached
e) L1 + L2
Spring 1 supports the weight.
Spring 2 supports the weight.
Both feel the same force, and
stretch the same distance as before.
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Recall -- Instantaneous acceleration
Velocity vector is always in the direction of motion;
acceleration vector can points in the direction
velocity is changing:
12
Circular Motion and Centripetal Force
An object moving in a circle must have a force acting
on it; otherwise it would move in a straight line!
The net force must have a component
centripetal
pointing to the center of the circle
force
This force may be provided by the tension in a
string, the normal force, or friction, or....
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Circular Motion and Centripetal Acceleration
An object moving in a circle must have a force acting
on it; otherwise it would move in a straight line.
a
a
If the speed is constant, the
direction of the acceleration
(which is due to the net
force) is towards the center
of the circle.
The magnitude of this
centripetal component of the
force is given by:
For circular motion problems, it is often convenient to choose coordinate
axes with one pointing along the direction of this centripetal force
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Circular Motion
An object may be changing its speed as it moves
in a circle; in that case, there is a tangential
acceleration as well:
15
Missing Link
A Ping-Pong ball is shot into a
circular tube that is lying flat
(horizontal) on a tabletop. When
the Ping-Pong ball leaves the
track, which path will it follow?
d
e
a b c
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Missing Link
A Ping-Pong ball is shot into a
circular tube that is lying flat
(horizontal) on a tabletop. When
the Ping-Pong ball leaves the
track, which path will it follow?
d
e
a b c
Once the ball leaves the tube, there is no longer a
force to keep it going in a circle. Therefore, it
simply continues in a straight line, as Newton’s
First Law requires!
Follow-up: What physical force provides the centripetal force?
17
Tetherball
a) toward the top of the pole
In the game of tetherball,
b) toward the ground
the struck ball whirls
c) along the horizontal component of the
tension force
around a pole. In what
direction does the net
d) along the vertical component of the
tension force
force on the ball point?
e) tangential to the circle
T
W
18
Tetherball
In the game of tetherball,
a) toward the top of the pole
the struck ball whirls
b) toward the ground
around a pole. In what
c) along the horizontal component of the
tension force
direction does the net
force on the ball point?
d) along the vertical component of the
tension force
e) tangential to the circle
The vertical component of the tension
balances the weight. The horizontal
component of tension provides the
W
T
T
necessary centripetal force that
accelerates the ball to towards the
pole and keeps it moving in a circle.
W
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20
Examples of centripetal force
when
no friction is
needed to hold
the track!
21
A hockey puck of mass m is attached to a string that passes
through a hole in the center of a table, as shown in the figure.
The hockey puck moves in a circle of radius r. Tied to the other
end of the string, and hanging vertically beneath the table, is a
mass M. Assuming the tabletop is perfectly smooth, what speed
must the hockey puck have if the mass M is to remain at rest?
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22
A hockey puck of mass m is attached to a string that passes
through a hole in the center of a table, as shown in the figure.
The hockey puck moves in a circle of radius r. Tied to the other
end of the string, and hanging vertically beneath the table, is a
mass M. Assuming the tabletop is perfectly smooth, what speed
must the hockey puck have if the mass M is to remain at rest?
necessary centripetal force:
Only force on puck is tension in the string!
To support mass M, the necessary tension is:
23
Circular motion and apparent weight
Car at the bottom of a dip
Note: at any specific
point, any curve can be
approximated by a portion
of a circle
This normal force is the
apparent, or perceived, weight
24
Going in Circles I
You’re on a Ferris wheel moving in a
vertical circle. When the Ferris wheel is
at rest, the normal force N exerted by
a) N remains equal to mg
b) N is smaller than mg
your seat is equal to your weight mg.
c) N is larger than mg
How does N change at the top of the
d) none of the above
Ferris wheel when you are in motion?
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Going in Circles I
You’re on a Ferris wheel moving in a
vertical circle. When the Ferris wheel is
at rest, the normal force N exerted by
a) N remains equal to mg
b) N is smaller than mg
your seat is equal to your weight mg.
c) N is larger than mg
How does N change at the top of the
d) none of the above
Ferris wheel when you are in motion?
You are in circular motion, so there
has to be a centripetal force pointing
inward. At the top, the only two
forces are mg (down) and N (up), so N
must be smaller than mg.
Follow-up: Where is N larger than mg?
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Vertical circular motion
Centripetal acceleration must be
A
vertical
(up)
B
horizontal
C
vertical
(down)
Condition for falling: N=0
at C:
So, as long as:
at the top, then N>0 and pointing down.
(note: before falling, apparent weight is
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in the opposite direction to true weight!)27
Chapter 7
Work and
Kinetic Energy
http://people.virginia.edu/~kdp2c/downloads/WorkEnergySelections.html
Work Done by a Constant Force
The definition of work, when the force is parallel to
the displacement:
SI unit: newton-meter (N·m) = joule, J
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29
Working Hard... or Hardly Working
Atlas holds up the
world.
Sisyphus pushes his
rock up a hill.
(b)
(a)
Who does more work?
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Working Hard... or Hardly Working
Atlas holds up the
world.
Sisyphus pushes his
rock up a hill.
(b)
(a)
With no displacement,
Atlas does no work
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Forces not along displacement
If the force is at an angle to the displacement:
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Friction and Work I
A box is being pulled
a) friction does no work at all
across a rough floor
b) friction does negative work
at a constant speed.
What can you say
c) friction does positive work
about the work done
by friction?
33
Friction and Work I
A box is being pulled
a) friction does no work at all
across a rough floor
b) friction does negative work
at a constant speed.
What can you say
c) friction does positive work
about the work done
by friction?
N Displacement
Friction acts in the opposite direction to
the displacement, so the work is negative.
Pull
f
Or using the definition of work (W = F
(Δr)cos ), because  = 180º, then W < 0.
mg
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Friction and Work II
Can friction ever do
positive work?
a) yes
b) no
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Friction and Work II
Can friction ever do
positive work?
a) yes
b) no
Consider the case of a box on the back of a pickup truck. If the box
moves along with the truck, then it is actually the force of friction that
is making the box move. ... but .... the friction isn’t really doing the
work, it is only transmitting the forces to the box, while work is done
by the truck engine.
My view: the language is confusing so
I’m not interested in arguing the point.
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Convenient notation: the dot product
The work can also be written as the dot product
of the force and the displacement:
vector “dot” operation: project
one vector onto the other
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Force and displacement
The work done may be positive, zero, or negative,
depending on the angle between the force and the
displacement:
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Sum of work by forces = work by sum of forces
If there is more than one force acting on an object,
the work done by each force is the same as the
work done by the net force:
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Units of Work
Lifting 0.5 L H2O up 20 cm = 1 J
40
Play Ball!
In a baseball game, the
catcher stops a 90-mph
a) catcher has done positive work
pitch. What can you say
b) catcher has done negative work
about the work done by the
c) catcher has done zero work
catcher on the ball?
41
Play Ball!
In a baseball game, the
catcher stops a 90-mph
a) catcher has done positive work
pitch. What can you say
b) catcher has done negative work
about the work done by the
c) catcher has done zero work
catcher on the ball?
The force exerted by the catcher is opposite in direction to the
displacement of the ball, so the work is negative. Or using the
definition of work (W = F (Δr)cos  ), because  = 180º, then W <
0. Note that the work done on the ball is negative, and its speed
decreases.
Follow-up: What about the work done by the ball on the catcher?
42
Tension and Work
A ball tied to a string is
a) tension does no work at all
being whirled around in
b) tension does negative work
a circle with constant
c) tension does positive work
speed. What can you
say about the work done
by tension?
43
Tension and Work
A ball tied to a string is
a) tension does no work at all
being whirled around in
b) tension does negative work
a circle with constant
c) tension does positive work
speed. What can you
say about the work done
by tension?
No work is done because the force
acts in a perpendicular direction to the
displacement.
Or using the
definition of work (W = F (Δr)cos  ),
T
 = 90º, then W = 0.
v
Follow-up: Is there a force in the direction of the velocity?
44
Work by gravity
Fg
A ball of mass m drops a distance h. What is the
total work done on the ball by gravity?
a
W = Fd = Fgx h
h
W = mgh
Path doesn’t matter when asking “how
much work did gravity do?” Only the
change in height!
A ball of mass m rolls down a ramp of height h at an
angle of 45o. What is the total work done on the ball by gravity?
a
Fgx = Fg sinθ
N
h = L sinθ
Fg
W = Fd = Fgx L = (Fg sinθ) (h / sinθ)
h
W = Fg h = mgh
θ
45
Motion and energy
When positive work is done on an object, its speed
increases; when negative work is done, its speed
decreases.
46
Kinetic Energy
As a useful word for the quantity of work we have
done on an object, thereby giving it motion, we
define the kinetic energy:
47
Work-Energy Theorem
Work-Energy Theorem: The total work done on an
object is equal to its change in kinetic energy.
(True for rigid bodies that remain intact)
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Lifting a Book
You lift a book with your hand in
a) mg  r
such a way that it moves up at
b) FHAND  r
constant speed. While it is
c) (FHAND + mg)  r
moving, what is the total work
d) zero
done on the book?
e) none of the above
r
FHAND
v = const
a=0
mg
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Lifting a Book
You lift a book with your hand in
a) mg  r
such a way that it moves up at
b) FHAND  r
constant speed. While it is
c) (FHAND + mg)  r
moving, what is the total work
d) zero
done on the book?
e) none of the above
The total work is zero because the net force
acting on the book is zero. The work done by
r
FHAND
the hand is positive, and the work done by
gravity is negative. The sum of the two is
v = const
a=0
zero. Note that the kinetic energy of the book
does not change either!
mg
Follow-up: What would happen if FHAND were greater than mg?
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Kinetic Energy I
By what factor does the
a) no change at all
kinetic energy of a car
b) factor of 3
change when its speed is
c) factor of 6
tripled?
d) factor of 9
e) factor of 12
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Kinetic Energy I
By what factor does the
a) no change at all
kinetic energy of a car
b) factor of 3
change when its speed is
c) factor of 6
tripled?
d) factor of 9
e) factor of 12
Because the kinetic energy is
mv2, if the speed increases by
a factor of 3, then the KE will increase by a factor of 9.
52
Slowing Down
If a car traveling 60 km/hr can
a) 20 m
brake to a stop within 20 m, what
b) 30 m
is its stopping distance if it is
c) 40 m
traveling 120 km/hr? Assume
d) 60 m
that the braking force is the same
e) 80 m
in both cases.
53 53
Slowing Down
If a car traveling 60 km/hr can
a) 20 m
brake to a stop within 20 m, what
b) 30 m
is its stopping distance if it is
c) 40 m
traveling 120 km/hr? Assume
d) 60 m
that the braking force is the same
e) 80 m
in both cases.
F d = Wnet = KE = 0 –
and thus, |F| d =
mv2,
mv2.
Therefore, if the speed doubles,
the stopping distance gets four
times larger.
54 54
Work Done by a Variable Force
We can interpret the work done graphically:
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Work Done by a Variable Force
If the force takes on several successive constant
values:
56
Work Done by a Variable Force
We can then approximate a continuously varying
force by a succession of constant values.
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Work Done by a Variable Force
The force needed to stretch a spring an amount x is
F = kx.
Therefore, the work
done in stretching the
spring is
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