Download Circular and Centripetal Motion

Circular and Centripetal Motion
• Moving objects can be described by using
kinematic equations.
• The motion of moving objects can be
explained by Newton’s Laws
• These principles can be applied to circular
motion as well.
Remember Newton’s 1st Law?
• An object at rest tends to stay at rest while an
object in motion tends to stay in motion at
constant velocity…
…unless acted on by an outside force.
What about circular motion?
Circular motion in not natural. A force is
required to change direction. A change in
direction means a change in velocity…
which means there is an acceleration.
The acceleration comes from the applied
force… uniform circular motion is caused by
an applied force.
Uniform Circular Motion
• Motion of an object in a circular pattern with
constant velocity
• It is only one of the forms of circular motion.
• An object moving in a circle will cover the same
linear distance each second.
• Example: a car moving in a circle at a constant
speed of 5 m/s will traverse 5 meters around the
perimeter per second.
• Circumference: distance of one complete cycle
around the perimeter
•
•
•
•
Average speed = distance/time
Average speed = circumference/time
Circumference = 2 x π x R / T
Period: one cycle around the circle
• Average speed = 2πR/ T
• R = radius of the circle; T = period
http://www.animations.physics.unsw.edu.au/jw/circular.htm#accelerat
ion
• http://www.upscale.utoronto.ca/PVB/Harrison/Fl
ash/ClassMechanics/RTZCoordSystem/RTZCoordS
ystem.html
• For objects moving around a circle with different
radii, in the same period, the object travelling the
greatest radius has the greatest speed.
• The average speed and the radius are directly
proportional.
• If the radius doubles, the speed will double.
The Velocity Vector
• If all objects moving in a circular motion have a
constant speed, will they have a constant
velocity?
• Review: vector vs. scalar
• The magnitude of the velocity vector is the
instantaneous speed of the object. The direction
of the vector is in the same direction as the
object moves.
• Since the object is moving in a circle, its direction
is constantly changing.
v
v
• The magnitude of the
vector is the same but the
direction is constantly
changing with position.
v
v
A better way to describe the direction of the
velocity vector is tangential. At any given instant
its direction is in the same direction as a tangent
drawn to the circle.
Think about acceleration…..
• If an object moving in uniform circular motion,
is there acceleration?
• An accelerating object is one that is changing
its velocity. Velocity is a vector with
magnitude (speed) and direction so a change
in either results in a change in velocity. In this
case, direction is changing so there is a change
in velocity and therefore acceleration.
Formula review…..
• Acceleration = change in velocity /change in
time
Or
a = ∆v / t
∆= vfinal – vinitial
Vi
vf
In the case of circular motion, the acceleration is directed
toward the center of the circle.
According to Newton’s second law, F = ma, an object that
is accelerating but be experiencing a net force.
The direction of the net force is in the same direction as
the acceleration. Therefore for an object moving in a
circle there must be an inward force acting on the
object to cause inward acceleration. This is known as
the centripetal force requirement. (means toward the
center or center seeking)
Note: Centripetal is not the same as centrifugal.
Centripetal vs. Centrifugal
• Centripetal force: the force that keeps the
object moving in a circle
• Centrifugal force: the force you “feel is being
exerted on you” … doesn’t actually exist…
– How you feel in a turning car
– How you feel at the top of a roller coaster
Review Newton’s First…
• The Law of Inertia:
– An object at rest remains at rest and an object in
motion will remain in motion at the same speed
and direction unless acted upon by an unbalanced
force.
– Moving objects tend to move in straight lines.
Therefore for an object to move in a circle, there
must be an unbalanced force present.
• Consider riding in a car or on a roller coaster.
The car begins to turn to the right. What
happens to the passenger (blue) in the car as
the driver (red) goes around a right curve or
makes a right turn? What happens to the
driver?
http://www.physicsclassroom.com/mmedia/ci
rcmot/rht.cfm
Examples:
1. A car turning a corner
– Force provided by friction between tires and
road
As a car makes a turn, the force of friction acting upon
the turned wheels of the car provides centripetal force
required for circular motion.
As a bucket of water is tied to a string and spun in a
circle, the tension force acting upon the bucket
provides the centripetal force required for circular
motion.
As the moon orbits the Earth, the force of gravity
acting upon the moon provides the centripetal force
required for circular motion.
Examples:
A ball twirled at the end of a string
– Force provided by tension in string
Tetherball
Examples:
Water that stays in a swinging bucket
– Force caused by the bottom of the bucket, or
the normal force
Water in glass Demo
Examples:
Earth orbiting around the sun
– Force provided by gravitational attraction
between two bodies
Solar System
What is the difference in linear and
centripetal forces?
Forces that cause things to speed up
we will call linear forces.
Forces that cause things to change
direction we will call centripetal
forces.
Can there be both?
Sure… a car can speed up around a corner, or
slow down as it turns.
Let’s draw a representation of the vectors
involved
a
F
r
V
m
Uniform Circular Motion
For uniform circular motion, the velocity is tangential to
the circle and perpendicular to the acceleration
Uniform Circular Motion
For uniform circular motion, the velocity is tangential to
the circle and perpendicular to the acceleration
• Period: the time it takes for an object to complete
one cycle, in this case a rotation, T
• Frequency: the number of times an object completes
a cycle in a given amount of time i.e. seconds
1
f
T
• Hertz: the unit to measure frequency instead of
saying s^-1 … 1/s = Hz
After closing a deal with a client, Kent leans back
in his swivel chair and spins around with a
frequency of 0.5 Hz. What is Kent’s period of
spin?
• Rotational speed: # of degrees or radians an
object in circular motion goes through per
second
• Linear speed: speed of an object in one
direction
• Tangential speed: the speed of an object at
any point on the edge of the circle (linear
speed for circular motion)
What equation can we use to measure the
speed of an object as it travels in circular
motion?
-How far does something travel to get all they way around
a circle?
-What do we call the time it takes to go that far?
Speed = Δ distance / Δ time
The time for an object to complete one revolution is
its period T…
The distance traveled in one revolution is
2 r
r
The speed of an object would then be…
2r
V
T
Curtis’ favorite record has a scratch 12 cm from
the center that makes the record skip 45 times
each minute. What is the linear speed of the
scratch as it turns?
Let’s look at what’s happening
mathematically…
Uniform Circular Motion
Newton’s 2nd Law: The net force on a body is equal to the product of the mass of the body
and the acceleration of the body.
F  ma
*The centripetal acceleration is caused by a centripetal
force that is directed towards the center of the circle.
F  ma  m
2
v
r
Equations
Fc  mac
F  ma  m
2
v
ac 
r
2
v
r
Now lets add these two…
2r
V
T
2
v
ac 
r
Lets try one together…
• A 13 g rubber stopper is attached to a 0.93 m string.
The stopper is swung in a horizontal circle, making
one revolution in 1.18 s. Find the tension force
exerted by the string on the stopper.
What if…
…the mass is doubled?
…the radius is doubled?
…the period is halved?
Missy’s favorite ride at the Topsfield Fair is the rotor, which has a radius of 4.0
m. The ride takes 2.0 s to make one full revolution. A) What is Missy’s linear
speed on the rotor? B) What is Missy’s centripetal acceleration on the rotor?
As their booster rockets separate, space shuttle astronauts typically feel accelerations
up to 3g, where g = 9.80 m/s2. In their training, astronauts ride in a device where they
experience such an acceleration as a centripetal acceleration. Specifically, the
astronaut is fastened securely at the end of a mechanical arm, which then turns at a
constant speed in a horizontal circle. Determine the rotation rate, in revolutions per
second, required to give an astronaut a centripetal acceleration of 3.00g while in
circular motion with radius 9.45 m.
An object of mass 0.500 kg is attached to the end of a cord whose
length is 1.50 m. The object is whirled in a horizontal circle. If
the cord can withstand a maximum tension of 50.0N, what is the
maximum speed the object can have before the cord breaks?
Lets try a problem or two, or three……
• The Texas Star Ferris wheel has a radius of 32
meters. When operating with constant
tangential velocity, it completes one rotation
in 2 minutes. A 60 kg rider sits on the bench
in one of the wheel’s baskets.
• What is the rider’s centripetal acceleration?
How big is the centripetal force required to
keep the rider going in the big circle at that
speed?