Download Chapter 7: Circular Motion and Gravitation

Chapter 7:
Circular Motion and Gravitation
Coach Kelsoe
Physics
Pages 233–267
Section 7–1:
Circular Motion
Coach Kelsoe
Physics
Pages 234–239
Section 7–1 Objectives:
• Solve problems involving centripetal
acceleration.
• Solve problems involving centripetal force.
• Explain how the apparent existence of an
outward force in circular motion can be
explained as inertia resisting the
centripetal force.
Centripetal Acceleration
• The cars on a rotating
Ferris wheel are said to
be in circular motion.
• Any object that revolves
about a single axis
undergoes circular
motion.
Tangential Speed
• The line about which
the rotation occurs is
called the axis of
rotation.
• Tangential speed (vt)
can be used to describe
the speed of an object
in circular motion.
Tangential Speed
• The tangential speed of a car on the Ferris
wheel is the car’s speed along an
imaginary line drawn tangent to the car’s
circular path. This definition can be
applied to any object moving in circular
motion.
• When the tangential speed is constant, the
motion is described as uniform circular
motion.
Tangential Speed
• Tangential speed depends on the distance
from the object to the center of the
circular path.
• To understand this concept, imagine a
carousel. The horses or carts on a
carousel are staggered so that some are
on the outside edge while some are closer
to the middle.
Tangential Speed
• Each horse/cart
completes one
circle in the
same time
period, but the
outside ones
cover more area,
therefore must
have a greater
tangential speed.
Centripetal Acceleration
• If the cart on a Ferris wheel is moving at a
constant speed around the wheel, it still
has an acceleration.
• Even though we typically think of
acceleration being a change of speed, it
can also be a change of direction.
• On a Ferris wheel, the carts are constantly
changing direction.
Centripetal Acceleration
• An acceleration of this nature is called a
centripetal acceleration – the
acceleration directed toward the center of
a circular path.
• The equation for finding the magnitude of
centripetal acceleration is given below:
ac = vt
2/r
Centripetal Acceleration
• Since acceleration is a vector quantity, we need
to know the direction of the acceleration. But if
direction constantly changes, how can we
accurately define the direction?
• Centripetal acceleration is always toward the
center of the circle! The word “centripetal”
actually means “center seeking.”
• We can better understand this idea by drawing
tangent vector lines or by adding opposite
vectors at two points.
Sample Problem A
• A test car moves at a
constant speed around
a circular track. If the
car is 48.2 m from the
track’s center and has
a centripetal
acceleration of 8.05
m/s2, what is the car’s
tangential speed?
Sample Problem A
• Given:
– r = 48.2 m
– ac = 8.05 m/s2
• Unknown
– vt = ?
• Solve
– ac = vt2/r , so…
– vt = √acr = √(8.05 m/s2)(48.2 m)
– vt = 19.7 m/s
Tangential Acceleration
• Centripetal acceleration results from a change in
direction, not a change in speed.
• In circular motion, an acceleration due to a
change in speed is called tangential acceleration.
• The easiest way to think of this is a car on a
circular track – it has centripetal acceleration no
matter what, due to its change in direction. It
will only have tangential acceleration if it speeds
up or slows down.
Centripetal Force
• Consider a ball of mass m
r
m
that is tied to a string of a
length r and that is being
whirled in a horizontal
circular path.
• Assume the ball moves
with a constant speed.
Centripetal Force
• Assume that the ball moves with
constant speed. Because the
velocity vector, v, continuously
changes direction during the
motion, the ball experiences a
centripetal acceleration that is
directed toward the center of the
motion.
• The inertia of the ball tends to
maintain the ball’s motion in a
straight path. However, the string
exerts a force that overcomes this
tendency.
r
m
Centripetal Force
• The net force that is directed toward the
center of an object’s circular path is called
centripetal force.
• Newton’s second law can be applied to find
the magnitude of this force: Fc = mac.
• The equation for centripetal acceleration
can be combined with Newton’s second
law to obtain the following equation:
Fc = mvt2/r
Centripetal Force
Centripetal Force
• Centripetal force is simply the name given
to the net force on an object in uniform
circular motion. Any type of force or
combination of forces can provide this net
force.
– Example: Friction between a race car’s tires
and a circular track is a centripetal force that
keeps the car in a circular path.
– Example: Gravitational force is a centripetal
force that keeps the moon in its orbit.
Centripetal Force
• Because centripetal force
acts at right angles to an
object’s circular motion, the
force changes the direction
of the object’s velocity. If
this force vanishes, the
object stops moving in a
circular path and instead,
moves along a straight line
path that is tangent to the
circle.
Rotating Systems
• Think about the feelings you experience as you
made a sharp turn in your vehicle. If you make
a sharp turn to your right, you are thrown
against the door to the left.
• If centripetal force is always toward the center of
the circular path, why wouldn’t you be thrown
toward the inside of the car rather than the
outside?
• A popular explanation is that a force must push
you outward. Many times this force is called
“centrifugal force,” but to lessen confusion, we
will refrain from using this term.
Rotating Systems
• So let’s explain the “false force” that is
centrifugal force.
• Before you begin to make your turn, your
body is following a straight-line path. As
the car enters the turn, your inertia makes
to tend to move along the original straightline path. This movement is in accordance
with Newton’s first law, which states that
the natural tendency of a body is to
continue moving in a straight line.
Rotating Systems
• If a large centripetal force acts on you, you
will move in the same direction as the car.
The origin of this force is the force of
friction between you and the car seat.
– Think about it: if your seat was slippery, and
the door wasn’t there, you’d slide right out!
• This gives you a great reason not to
Armor-All your seats in a Jeep with no
doors!
Inertia, not Centrifugal Force!
Simulated Gravity Using
Centripetal Force!
• http://www.courses.psu.edu/aersp/aersp0
55_r81/station/station.html