Transcript Circular Motion

Circular Motion
• Uniform Circular Motion is rotational motion
where the angular velocity of an object is
constant.
• Because we are moving in circles and
changing the direction of the velocity vector,
there must be an acceleration present, even
though our speed remains constant. We will
discuss this further, later in this unit.
• The key to understanding all types of circular
motion (uniform and accelerated) is to
understand that all of the circular variables
have the same relationships as their linear
counterparts.
Understanding angles:
Degrees and Radians
• Degrees and Radians are two types of
angular measurement.
• 1 rad = approx. 57.3o
• 1o = approx. 1.74 x 10-2 rad
• To convert, set up a ratio where x o
y radians
o
360

2 radians
• These are approximations because of the
 factor in the conversion.
Angular displacement
• Roughly, displacement refers to how far
something moves from its starting
position.
• Since revolutions in a circle or on a
wheel would continue to bring a
reference point on the circle back to the
same spot (x=0) angular displacement
refers to the quantity of angles the
reference point has swung through.
• The symbol theta () is used for angular
displacement.
Sometimes this is
also called “s,” the
“arc length.”
Angular velocity: How fast something
spins.
• Linear velocity is equal to change in
displacement over change in time.
Unfortunately, linear velocity, as it
relates to displacement is not terribly
meaningful when describing movement
in circles.
• Why do you suppose this is?
• Angular velocity describes how fast
something spins and thus must be
related to angular displacement.
• Angular velocity is equal to the net
change in angular position divided by
the time taken making that change.
•  = /t
Angular acceleration
• Angular acceleration, like linear
acceleration, refers to how fast a velocity
changes.
• Angular acceleration is equal to the
change in angular velocity over time.
•  = /t
Linear and Angular Motion
Quantity Linear Angular Relationships
Symbol Symbol
Displacement
x
x = r

Velocity
v

v = r
Acceleration
a

at = r
What’s up with this “t” going on here? This refers to the
tangential component of the acceleration. That is to say
the acceleration that acts around the circle.
Relationships:
 = /t
 = /t
 = o + t
 = o +t
 = ot + ½t2
2 - o2 = 2
Look familiar?
Now it gets icky...
• The period of revolution is the amount
of time it takes the rotating object to
make one complete revolution.
2r
T
v
When you know the
linear components.
T
2

When you know the
angular components.
Centripetal Acceleration
• A rotating body is in many respects like a projectile
(a projectile with a high enough horizontal velocity
is a satellite, an object that orbits, or rotates
around the earth).
• Rotating bodies have a component of acceleration
that constantly pulls the object along the radius
towards the center of the circle or orbit. This is the
centripetal acceleration.
• Centripetal acceleration, since it is perpendicular to
the velocity vector at all times, does not change
the magnitude of said vector, only the direction.
The concept is similar to the change in direction
that a projectile undergoes during flight.
The centripetal acceleration,
ac, ensure that the object continues
to “fall”towards the center
of the circle as it
rotates.
at  r
Tangential
2
v
ac 
  2r
r
Angular
The tangential
acceleration, at,
and the tangential velocity
and displacement relate respectively to the object’s acceleration,
velocity and displacement around the circle.
Centripetal Force
Centripetal force is simply to force associated with and
causing the centripetal acceleration, ac.
2
mv
F  ma
r
Any object traveling in a circle or a circular path
experiences a centripetal force.
The car problem:
You are in the backseat of your parents sedan.
When the car turns sharply to the right (as in making a 90o turn at a
corner) your body is forced to the left. If a centripetal force is
actually pulling you and the car radially towards the inside of the
turn, why do you feel a force pushing you away from the center of
rotation? (hint: think about Newton’s 1st Law!)
Centripetal Force:
A NET Force
• The centripetal force is, in physics, what we call an
imaginary force. This does NOT mean that the
force is non-existent, but rather this means that
the force we identify mathematically as
“centripetal” is ALWAYS caused by some other
force.
– E.g., a contact (normal) force, a friction or a tension.
• This means that mv2/r is always equal to
something else that you must define based on the
circumstances of the system you are studying.
Torque
• Torque is simply a force applied to an
object at some radial distance from the
center of rotation. If the torque is large
enough to overcome the rotational
inertia of the object it will cause the
object to rotate.
• The “radial distance” is sometimes
called the lever arm.
• Torque, like force, is governed by
Newton’s Second Law:   
Force
Radius
    Fr
Torque: Lower case “tau”
Angular acceleration
Moment of inertia:
describes the arrangement
of mass around a central point,
often something you look up...
Torques work in equilibrium just like forces,
and all of the same rules apply. E.g., When the net torque
on an object is zero it is either resting, or rotating with
constant rotational velocity.
    Fr
The torque used to create or sustain a rotation is equal to
the applied force multiplied by the radius at which that
force is applied if, and only if, the force is applied such
that is is perpendicular to the radius.
We will make this assumption in the course when solving
problems unless otherwise stated.
Some problems for your mental stimulation
pleasure...
1) A wheel rotates at a rate of 8 rad/s for 16 s. What is the angular
displacement of the wheel? How many revolutions as it made?
2) If the wheel in problem 1 is actually a car tire with a diameter of
0.72 m (roughly 32 inches), how far has the car traveled?
3) a) Calculate the centripetal acceleration on a roller coaster car of mass m =
100 kg that goes through a circular loop of radius 25m at 55 m/s.
b) what is the centripetal force on the car and occupants?
c) what is the angular velocity of the roller coaster?
4. A 0.50 m crowbar is used to rip open a wooden crate. What force is
necessary to apply to the crowbar if the torque necessary to strip boards off the
crate is 47 Nm?
5) If the angular acceleration caused by the torque in problem 4 is 0.2 rad/s2,
what is the moment of inertia for the crowbar?