LECTURE 11
CIRCULAR MOTION
Instructor: Kazumi Tolich
Lecture 11
2
¨
Reading chapter 6-5
¤
¤
¤
Uniform circular motion
Centripetal and tangential acceleration of non-uniform circular motion
Dynamics of circular motion
n
n
n
Centripetal force
Unbanked and banked curves
Loop-the-Loop
Centripetal force and acceleration
3
¨
¨
¨
Centripetal force, 𝐹"# , is any form of force (normal force,
tension, gravity, etc) applied toward the center of the circle
that keeps an object in a circular motion.
Centripetal acceleration, 𝑎"# , is an acceleration of an
object caused by a centripetal force.
The magnitude of a net centripetal force required for an
object with a mass 𝑚 going around in a circular path with a
radius 𝑟 with a uniform speed 𝑣 is given by
( 𝐹"# = 𝑚𝑎"#
𝑣+
=𝑚
𝑟
Velocity and acceleration of a circular motion
4
¨
The direction of the velocity is always tangential to the path, perpendicular to the circle’s radius.
¨
An object under a uniform circular motion moves at constant speed and has a centripetal acceleration.
¨
An object in a circular motion with varying speed has both centripetal and tangential accelerations.
𝐚-.-/0 = 𝐚"# + 𝐚-
Quiz: 1
¨
You are in the front passenger seat of a car. While the car makes a sharp left turn, you found
yourself leaning toward the passenger door (to your right). Which of the following statements
is/are correct? Choose all that apply.
A.
B.
C.
D.
Centrifugal force is pushing you to the right.
Centripetal force is pushing you to the right.
The seat is exerting a leftward frictional force on you.
The seat is exerting a rightward frictional force on you.
Quiz: 11-1 answer
¨
¨
¨
¨
¨
The seat is exerting a leftward frictional force on you.
For you to make the turn, there must be centripetal force acting on you toward the
center of the curvature.
The static frictional force by the seat on your bottom is supplying that centripetal
force.
Meanwhile, your head tends to keep going in a straight path due to its inertia.
This fictitious “force” to the right is sometimes called “centrifugal force.” But there is
no such force.
Why don’t satellites fall into Earth?
7
Q.
A.
Why don’t satellites fall into Earth because of Earth’s
gravity?
They do.
The tangential velocity of the satellite is fast enough so
that the distance the satellite falls towards Earth, and the
distance it travels in the tangential direction in a given time
follow the satellite’s orbit.
Example: 1
8
¨
Igor is an engineer in a spacecraft orbiting
Earth at an altitude ℎ = 520 km with a
constant speed 𝑣 = 7.6 km/s. Igor’s mass is
𝑚 = 79 kg.
a)
b)
What is his acceleration?
What (centripetal) gravitational force does
Earth exert on Igor?
Quiz: 2
9
¨
You drive your dad’s car too fast around a curve, and the car starts to skid.
What is the correct description of this situation? Choose all that apply.
A.
The car’s engine is not strong enough to keep the car from being pushed out.
B.
Static friction between the tires and the road is not strong enough to keep the car in the
circle.
C.
Kinetic friction between the tires and the road is not strong enough to keep the car in the
circle.
D.
The car is too massive to make the turn.
Quiz: 11-2 answer
10
¨
¨
¨
¨
¨
¨
Static friction between the tires and the road is not strong enough to keep the car in the circle.
On a unbanked road, static friction by the road on the car provides the centripetal force necessary for the
car to follow a curve.
It is static friction, not kinetic friction, because no slipping occurs at the point of contact between road and
tires.
The maximum speed the car can go without skidding is limited by the maximum static friction the road can
exert on the tire.
The maximum speed is independent of mass of the car.
The car’s engine is internal to the car, so it does not provide centripetal force to the car.
Example: 2
11
A bicyclist travels at a constant
speed of 𝑣 = 9.00 m/s in a circle
of radius 𝑟 = 25.0 m on a flat
ground. The combined mass of the
bicycle and rider is 𝑚 = 85.0 kg.
¨
a)
b)
Calculate the magnitude of the force
of friction exerted by the road on the
bicycle.
If the coefficient of static friction
between the tires and this road is
𝜇4 = 1.0, what is the maximum speed
the bicycle can go before skidding?
Quiz: 3
12
¨
A car is following a banked curve without skidding. The road is
very icy, so it is not providing any frictional force. What is
providing the centripetal force? Choose all that apply.
A.
B.
C.
D.
Normal force by the road
Weight of the car
Actually, this is not possible. Friction is always needed.
Actually, centripetal force is not necessary.
Quiz: 11-3 answer
13
¨
¨
¨
On a banked road, the normal force
by the road will have a component in
the centripetal direction.
The banking angle 𝜃 is usually
chosen so that no friction is needed
for a car to complete the curve at
the specified speed.
Banking angle is given by
𝑣+
tan 𝜃 =
𝑟𝑔
Quiz: 4
14
¨
You are riding a bicycle and comfortably making around a loop-theloop. Which of the following forces are providing the centripetal force
at the top of the loop-the-loop? Choose all that apply.
A.
B.
C.
D.
Weight of you and bicycle
Normal force by the track
Frictional force by the track
Nothing. Centripetal force is not needed.
Quiz: 11-4 answer
15
¨
¨
¨
The combination of weight of you and bicycle and
normal force by the track is providing the centripetal
force.
If you are making the loop-the-loop, the bike is
touching the track, so the normal force on the bike is
non-zero and pointing down, towards the center of the
loop.
Weight is also downward towards the center of the
loop.
Example: 3
16
¨
The radius of curvature of the track at the top of
a loop-the-loop on a roller-coaster ride is 𝑟. The
mass of the roller-coaster is 𝑚.
a)
b)
¨
Calculate the speed of the coaster, 𝑣, as a function
of the normal force on the coaster by the track, 𝑁;< .
What is the minimum speed at which the coaster
must be going at the top of the loop-the-loop to
barely make it?
Loop-the-loop fails when you do not have enough
speed at the top.
¤
http://www.youtube.com/watch?v=tzQJNeqiGG4