Circular Motion
Lecturer:
Professor Stephen T. Thornton
Reading Quiz: When a car is moving
around a banked curve, which of the
following statements is most true?
A) The friction is always up the slope of the
banked road.
B) The friction is always down the slope of
the banked road.
C) The friction can either be up or down the
slope of the banked road.
D) It sounds as if the car’s tires are too bald
to matter.
Answer: C
Friction can be up
or down the road.
This is in many
textbooks and is
worked out. There
is even one speed
where there is no
friction.
Last Time
Review
Friction
Today
Circular motion
Centripetal motion, force
Motion on banked curves
Try this with
a block of
wood. This is
similar to the
car moving
on banked
curve.
Conceptual Quiz
Below you see two cases:
a father pulling or
pushing a sled with a
force F that is applied at
an angle q. In which case
is the normal force
greater?
A)
B)
C)
D)
case 1
case 2
it’s the same for both
depends on the magnitude of
the force F
E) depends on the ice surface
Case 1
Case 2
Conceptual Quiz
Below you see two cases:
a father pulling or
pushing a sled with a
force F that is applied at
an angle q. In which case
is the normal force
greater?
A)
B)
C)
D)
case 1
case 2
it’s the same for both
depends on the magnitude of
the force F
E) depends on the ice surface
Case 1
In case 1, the force F is pushing
down (in addition to mg), so the
normal force needs to be larger. In
case 2, the force F is pulling up,
against gravity, so the normal
force is lessened.
Case 2
Conceptual Quiz
Two blocks of masses 2m
and m are in contact on a
horizontal frictionless
surface. If a force F is
applied to mass 2m, what
is the force on mass m ?
The blocks accelerate.
A) 2F
B) F
C)
1
2
F
D)
1
3
F
E)
1
4
F
F
2m
m
Conceptual Quiz
Two blocks of masses 2m
and m are in contact on a
horizontal frictionless
surface. If a force F is
applied to mass 2m, what is
the force on mass m ? The
blocks accelerate.
The force F leads to a specific
acceleration of the entire system. In
order for mass m to accelerate at the
same rate, the force on it must be
smaller! How small?? Let’s see...
A) 2F
B) F
C)
1
2
F
D)
1
3
F
E)
1
4
F
F
F = 3ma; a = F/3m, F1=ma=F/3
Follow-up: What is the acceleration of each mass?
2m
m
Conceptual Quiz
on a frictionless truck bed.
A) the force from the rushing air
pushed it off
When the truck accelerates
B) the force of friction pushed it off
forward, the box slides off
C) no net force acted on the box
the back of the truck
D) truck went into reverse by accident
A box sits in a pickup truck
because:
E) none of the above
Conceptual Quiz
on a frictionless truck bed.
A) the force from the rushing air
pushed it off
When the truck accelerates
B) the force of friction pushed it off
forward, the box slides off
C) no net force acted on the box
the back of the truck
D) truck went into reverse by accident
A box sits in a pickup truck
because:
E) none of the above
Generally, the reason that the box in the truck bed would move
with the truck is due to friction between the box and the bed.
If there is no friction, there is no force to push the box along,
and it remains at rest. The truck accelerated away, essentially
leaving the box behind!!
Mass Moving Up Ramp. A small block
of mass m is given an initial speed up a
ramp inclined at angle q to the horizontal.
It travels a distance d up the ramp and
comes to rest. (a) Determine a formula for
the coefficient of kinetic friction between
block and ramp. (b) What can you say
about the value of the coefficient of static
friction?
Circular motion
Do demo with
string and ball.
Note that the
direction of the
velocity is
changing. The
ball is
accelerating!
v  v f  vi
Notice that v tends to
point towards the center of
the circle. As q becomes
smaller and smaller, v
points directly to center.
Therefore the acceleration
points towards the center
of the circle.
Centripetal acceleration
Centripetal means “center seeking”.
v v2  v1
aav 

t
t
Look at your textbook for a derivation
of the magnitude of the centripetal
acceleration acp :
v2
acp 
r
where r is the radius and v is the speed.
v
a
r
Dynamics of Uniform Circular Motion
We can see that the
force must be inward
by thinking about a
ball on a string.
Strings only pull;
they never push.
For an object to be in uniform circular motion,
there is an acceleration, and, therefore, a net
force acting on it.
We already know the
acceleration, so we can
immediately write the
force:
å
2
v
FR = maR = macp = m
r
Centripetal force
Where in the world did this centripetal
force come from?
There has to be a force to keep the
object moving in a circle. In the case
of the ball and string, it is the tension in
the string. The tension always points
towards the center!
The direction of the centripetal force
must also be towards the center!
The moon rotates around the Earth in a circle.
What is the centripetal force that causes this?
If you drive around in a circle with a bicycle
or even with a car, what is the centripetal
force?
In a simple atomic model of the hydrogen
atom, the electron rotates around the proton in
a circle. What is the centripetal force?
How can we make a bowling ball go
around in a circle?
How are you going to do it?
What is the centripetal force?
Circular motion
Results for circular motion:
 Consider an object moving in a circle of
radius r with a constant speed v.
 A centripetal acceleration of magnitude
v2/r must be present.
 There must be a centripetal force Fcp of
value
mv 2
Fcp  macp 
r
Conceptual Quiz: What
other forces are exerted
on the ball besides mg?
A) Friction
B) Tension
C) A normal force
perpendicular to mg.
D) A normal force
perpendicular to the
surface of the cone at
the ball.
Answer: D
The only other
possible force is the
normal force, and it
must be perpendicular
to the surface that the
ball is rolling upon.
N
Conceptual Quiz: What is
the direction of the net
force?
A) towards the center of
the dashed circle at the
ball (radially).
B) away from the center
of the circle at the ball.
C) up at the ball.
D) down at the ball.
E) cannot tell with
information given.
Answer: A
Because the ball is
moving at constant
speed in a circle, the
net force must be
along the radial
direction, towards
the center of the
circle. This is the
centripetal force.
Look at ball moving in vertical circle
In this case we do not
have uniform circular
motion. The tension
always points towards
the center, but gravity
points down.
Look at Examples in
textbook.
Turning corners
What do we notice when we turn
corners at high speed?
 Good chance of falling
over or skidding.
 When skiing, we lean
over and tilt our skis.
 Interstate highways
are banked.
 Motorcycles tilt.
f
Highway Curves: Banked and Unbanked
When a car goes around a curve, there must be
a net force toward the center of the circle of
which the curve is an arc. If the road is flat, that
force is supplied by friction.
If the frictional force is
insufficient, the car will
tend to move more
nearly in a straight line,
as the skid marks show.
You really do not want your tires to slip!!
As long as the tires do not slip, the friction is
static. If the tires do start to slip, the friction is
kinetic, which is bad in two ways:
1. The kinetic frictional force is smaller than the
static.
2. The static frictional force can point toward the
center of the circle, but the kinetic frictional
force opposes the direction of motion, making
it very difficult to regain control of the car and
continue around the curve.
Banking the curve can help
keep cars from skidding. In
fact, for every banked curve,
there is one speed at which
the entire centripetal force is
supplied by the horizontal
component of the normal
force, and no friction is
required. This occurs when:
Whirling Bucket. A bucket of mass 2.00
kg is whirled in a vertical circle of radius
1.10 m. At the lowest point of its motion the
tension in the rope supporting the bucket is
25.0 N. (a) Find the speed of the bucket.
(b) How fast must the bucket move at the
top of the circle so that the rope does not go
slack?
Conceptual Quiz
Antilock brakes keep the
car wheels from locking
and skidding during a
sudden stop. Why does
this help slow the car
down?
A) mk > ms so sliding friction is better
B) mk > ms so static friction is better
C) ms > mk so sliding friction is better
D) ms > mk so static friction is better
E) none of the above
Conceptual Quiz
Antilock brakes keep the
car wheels from locking
and skidding during a
sudden stop. Why does
this help slow the car
down?
A) mk > ms so sliding friction is better
B) mk > ms so static friction is better
C) ms > mk so sliding friction is better
D) ms > mk so static friction is better
E) none of the above
Static friction is greater than sliding friction, so
by keeping the wheels from skidding, the static
friction force will help slow the car down more
efficiently than the sliding friction that occurs
during a skid.
Conceptual Quiz
A box sits on a flat board.
You lift one end of the
board, making an angle
with the floor. As you
increase the angle, the
box will eventually begin
to slide down. Why?
A) component of the gravity force
parallel to the plane increased
B) coefficient of static friction
decreased
C) normal force exerted by the board
decreased
D) both A and C
E) all of A, B, and C
Normal
Net Force
Weight
Conceptual Quiz
A box sits on a flat board.
You lift one end of the
board, making an angle
with the floor. As you
increase the angle, the
box will eventually begin
to slide down. Why?
A) component of the gravity force
parallel to the plane increased
B) coefficient of static friction
decreased
C) normal force exerted by the board
decreased
D) both A and C
E) all of A, B, and C
As the angle increases, the component of
weight parallel to the plane increases and
the component perpendicular to the plane
Normal
decreases (and so does the normal force).
Because friction depends on normal force,
Net Force
we see that the friction force gets smaller
and the force pulling the box down the
plane gets bigger.
Weight
Conceptual Quiz
A mass m is placed on an
inclined plane (m > 0) and
slides down the plane with
constant speed. If a similar
block (same m) of mass 2m
were placed on the same
incline, it would:
m
A) not move at all
B) slide a bit, slow down, then stop
C) accelerate down the incline
D) slide down at constant speed
E) slide up at constant speed
Conceptual Quiz
A mass m is placed on an
inclined plane (m > 0) and
slides down the plane with
constant speed. If a similar
block (same m) of mass 2m
were placed on the same
incline, it would:
A) not move at all
B) slide a bit, slow down, then stop
C) accelerate down the incline
D) slide down at constant speed
E) slide up at constant speed
The component of gravity acting down
N
f
the plane is double for 2m. However,
the normal force (and hence the friction
force) is also double (the same factor!).
This means the two forces still cancel
to give a net force of zero.
Wy
q
Wx
W
q