Drag Forces
Lecturer:
Professor Stephen T. Thornton
Reading Quiz
A skier goes over a small round hill
A) Fc = N + mg
with radius R. Because she is in
B) Fc = mg – N
circular motion, there has to be a
C) Fc = T + N – mg
centripetal force. At the top of the
D) Fc = N
hill, what is Fc of the skier equal to?
E) Fc = mg
v
R
Reading Quiz
A skier goes over a small round hill
A) Fc = N + mg
with radius R. Because she is in
B) Fc = mg – N
circular motion, there has to be a
C) Fc = T + N – mg
centripetal force. At the top of the
D) Fc = N
hill, what is Fc of the skier equal to?
Fc points toward the center of
the circle (i.e., downward in
this case). The weight vector
points down and the normal
force (exerted by the hill)
points up. The magnitude of
the net force, therefore, is
Fc = mg – N.
E) Fc = mg
v
mg
N
R
Follow-up: What happens when
the skier goes into a small dip?
Last Time
Circular motion
Motion on banked curves
Lots of Conceptual Quizzes
Today
Non-uniform circular motion
Drag
Terminal velocity
Fundamental forces
Banked Curve. A curve of radius 68 m is
banked for a design speed of 85 km/h. If the
coefficient of static friction is 0.30 (wet
pavement), at what range of speeds can a
car safely make the curve? [Hint: Consider
the direction of the friction force when the
car goes too slow or too fast.]
Conceptual Quiz
In the game of
tetherball, the struck
ball whirls around a
pole. In what
direction does the
net force on the ball
point?
A) toward the top of the pole
B) toward the ground
C) along the horizontal component
of the tension force
D) along the vertical component
of the tension force
E) tangential to the circle
T
W
Conceptual Quiz
In the game of
tetherball, the struck
ball whirls around a
pole. In what
direction does the
net force on the ball
point?
A) toward the top of the pole
B) toward the ground
C) along the horizontal component
of the tension force
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 centripetal force
that points toward the center
of the circle.
W
T
T
W
Conceptual Quiz
You drive your car too
fast around a curve and
the car starts to skid.
What is the correct
description of this
situation?
A) car’s engine is not strong
enough to keep the car from
being pushed out
B) friction between tires and road
is not strong enough to keep car
in a circle
C) car is too heavy to make the
turn
D) a deer caused you to skid
E) none of the above
Conceptual Quiz
You drive your car
too fast around a
curve and the car
starts to skid. What
is the correct
description of this
situation?
A) car’s engine is not strong enough
B)
C)
D)
E)
to keep the car from being
pushed out
friction between tires and road is
not strong enough to keep car in a
circle
car is too heavy to make the turn
a deer caused you to skid
none of the above
The friction force between tires
and road provides the
centripetal force that keeps the
car moving in a circle. If this
force is too small, the car
continues in a straight line!
Follow-up: What could be done to
the road or car to prevent skidding?
Conceptual Quiz
A Ping-Pong ball is shot
into a circular tube that is
lying flat (horizontal) on a
tabletop. When the PingPong ball leaves the track,
which path will it follow?
Conceptual Quiz
A Ping-Pong ball is shot
into a circular tube that is
lying flat (horizontal) on a
tabletop. When the PingPong ball leaves the track,
which path will it follow?
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?
Conceptual Quiz
You swing a ball at the end of string
A) Fc = T – mg
in a vertical circle. Because the ball
B) Fc = T + N – mg
is in circular motion there has to be a
C) Fc = T + mg
centripetal force. At the top of the
D) Fc = T
ball’s path, what is Fc equal to?
E) Fc = mg
top
v
R
Conceptual Quiz
You swing a ball at the end of string
A) Fc = T – mg
in a vertical circle. Because the ball
B) Fc = T + N – mg
is in circular motion there has to be a
C) Fc = T + mg
centripetal force. At the top of the
ball’s path, what is Fc equal to?
Fc points toward the center of the circle
(i.e., downward in this case). The
weight vector points down and the
D) Fc = T
E) Fc = mg
v
mg
T
tension (exerted by the string) also
points down. The magnitude of the
net force, therefore, is Fc = T+ mg.
R
Conceptual Quiz
A rider in a “barrel of
fun” finds herself stuck
with her back to the wall.
Which diagram correctly
shows the forces acting
on her?
A
B
C
D
E
Conceptual Quiz
A rider in a “barrel of fun”
finds herself stuck with
her back to the wall.
Which diagram correctly
shows the forces acting
on her?
A
B
C
The normal force of the wall on the
rider provides the centripetal force
needed to keep her going around
in a circle. The downward force of
gravity is balanced by the upward
frictional force on her, so she does
not slip vertically.
See http://physics.bu.edu/~duffy/semester1/semester1.html
D
E
Nonuniform Circular Motion
If an object is moving in a
circular path but at varying
speeds, it must have a
tangential component to its
acceleration as well as the
radial one.
Nonuniform Circular Motion
This concept can be used for an object
moving along any curved path, as any
small segment of the path will be
approximately circular.
Drag Forces
1
FD = r ACD v 2
2
Here,
FD is the drag force;
ρ the density of the
medium;
A the cross-sectional
area of the object;
CD the drag coefficient.
Velocity-Dependent Forces:
Drag and Terminal Velocity
When an object moves through a fluid at low
speed, it experiences a drag force that
depends on the velocity of the object.
FD  bv
For small velocities, the force is approximately
proportional to the velocity; for higher speeds,
the force is approximately proportional to the
square of the velocity.
2
FD
v
If the drag force on a falling
object is proportional to its
velocity, the object gradually
slows until the drag force and
the gravitational force are equal.
Then it falls with constant
velocity, called the terminal
velocity (~120 mph for humans).
mg  bv when equal, no net force
mg
vT 
terminal velocity
b
Simplified Top View of a
Centrifuge in Operation
More dense
Less dense
Separate red and
white blood cells
from serum.
anemia  45%
Centrifuges
Centrifuges have many uses.
• Separating red and white blood cells
from serum.
• Separating 235U from 238U to produce
fissionable material.
• Separating large molecules from
smaller molecules for research and
pharmaceutical purposes.
Fundamental Forces
• Universal gravitation: gravitational force
between any two masses
• Electroweak force: includes electric, magnetic,
and weak nuclear forces (responsible for
radioactive decay)
• Strong (nuclear) force: binds protons and
neutrons together in nucleus
Conceptual Quiz
A box of weight 100 N is at
rest on a floor where ms = 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
B) moves to the right
C) moves up
D) moves down
E) the box does not move
Static friction
(ms = 0.5 )
m
T
Conceptual Quiz
A box of weight 100 N is at
rest on a floor where ms = 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
B) moves to the right
C) moves up
D) moves down
E) the box does not move
The static friction force has a
maximum of msN = 40 N. The
tension in the rope is only 30 N.
Static friction
(ms = 0.4 )
m
T
So the pulling force is not big
enough to overcome friction.
Follow-up: What happens if the tension is 35 N? What about 45 N?