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
Why don’t satellites fall into Earth?
6
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
7
¨
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
8
Example: 2
9
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 & 4
10
Example: 3
11
¨
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