Circular Motion
September 29, 2016
Be able to convert radians to degrees to revolutions
Warm Up
An okapi is running in a circle with a speed of 200 per second.
Determine the speed in rad/s.
Determine the speed in rpms.
Objectives:
ð Know the properties of Circular Motion o Be able to convert radians to degrees to revolutions
§ Understand the difference between tangential components and radial components and be able to use equations the relate the two
o Be able to relate these words to circular motion, know their equations and their units
§ period, arc length, angular position, tangential velocity, angular velocity, centripetal acceleration, angular acceleration, tangential acceleration
o Be able to recognize when a problem is in UNIFORM circular motion or in NON­UNIFORM circular motion
§ Be able to use kinematics equations to solve problems § Be able to use calculus to solve problems with a changing angular acceleration
o Be able to graph angular position vs. time, angular velocity vs. time, and angular acceleration vs. time for a given scenario (or describe the motion of given graphs)
Circular Motion
September 29, 2016
Be able to relate these words to circular motion, know their equations and their units
period, arc length, angular position, tangential velocity, angular velocity, centripetal acceleration, angular acceleration, §
tangential acceleration
Uniform Circular Motion
Distance (arc length):
s = rθ
(θ in radians)
Tangential velocity:
vt = 2πr
T
vt = ds = r dθ
dt
dt
Angular velocity
Symbol: ω
(Greek letter lowercase omega)
-1
Units: rad/s = s
vt = rω
Be able to relate these words to circular motion, know their equations and their units
period, arc length, angular position, tangential velocity, angular velocity, centripetal acceleration, angular acceleration, §
tangential acceleration
Other properties:
• Counter-clockwise motion is positive for vt and ω
• Clockwise motion is negative for vt and ω
• Radians, degrees, rotations, and revolutions are
considered dimensionless (not really units), even though it
is often very helpful to have them there!
Circular Motion
September 29, 2016
­Know the properties of Circular Motion ­Understand the difference between tangential components and radial components and be able to use equations the relate the two
A 3000 m high mountain is located on the
equator. How much faster does a climber on
top of the mountain move than a surfer at a
nearby beach? The earth's radius is 6400 km.
Circular Motion
September 29, 2016
­Understand the difference between tangential components and radial components and be able to use equations the relate the two
­Be able to relate these words to circular motion, know their equations and their units
Acceleration
Find the direction of the
acceleration vectors.
­Understand the difference between tangential components and radial components and be able to use equations the relate the two
­Be able to relate these words to circular motion, know their equations and their units
We name this acceleration "centripetal acceleration"
because the word centripetal means "center
seeking." (Symbol: ac)
What is the "angular acceleration" for UCM? (Symbol: α)
Another type of acceleration associated with circular motion
is called "tangential acceleration." This vector always points
tangent to the circle. (Symbol: at)
What is the magnitude of "tangential acceleration" for UCM?
Circular Motion
September 29, 2016
­Understand the difference between tangential components and radial components and be able to use equations the relate the two
­Be able to relate these words to circular motion, know their equations and their units
Anyone recall the equation for centripetal acceleration?
Determine an equation for centripetal acceleration in terms of angular speed.
­Understand the difference between tangential components and radial components and be able to use equations the relate the two
­Be able to relate these words to circular motion, know their equations and their units
A typical carnival Ferris wheel has a radius
of 9.0 m and rotates 4.0 times per minute.
What magnitude acceleration do the riders
experience? What type of acceleration is
this?
Circular Motion
September 29, 2016
­Understand the difference between tangential components and radial components and be able to use equations the relate the two
­Be able to relate these words to circular motion, know their equations and their units
Rank in order, from largest to smallest, the centripetal
accelerations.
o
Be able to recognize when a problem is in UNIFORM circular motion or in NON­UNIFORM circular motion
Nonuniform Circular Motion and Angular Acceleration
Just as acceleration is the derivative of velocity,
angular acceleration is the derivative of angular velocity.
Symbol for angular acceleration:
What are the units for angular acceleration?
Circular Motion
September 29, 2016
o
Be able to recognize when a problem is in UNIFORM circular motion or in NON­UNIFORM circular motion
Determining + and ­
ω is positive when
ω is negative when
α is positive when
α is negative when
For these pictures, state the signs of ω and α
o
Be able to recognize when a problem is in UNIFORM circular motion or in NON­UNIFORM circular motion
Be able to use calculus to solve problems with a changing angular acceleration
§
The angular velocity of a rotating object is given by the equation
ω(t) = 3t2 + 4t + 6
Determine the angular displacement between t = 0 and t = 2 seconds.
Determine the equation for angular acceleration.
What is angular acceleration at t = 1 second?
Circular Motion
September 29, 2016
o
Be able to graph angular position vs. time, angular velocity vs. time, and angular acceleration vs. time for a given scenario (or describe the motion of given graphs)
Given the graph of angular velocity vs. time, sketch the corresponding angular acceleration vs. time graph. Label acceleration numbers.
Describe the motion.
α
t
o
Be able to recognize when a problem is in UNIFORM circular motion or in NON­UNIFORM circular motion
Be able to use kinematics equations to solve problems
§
If we have a CONSTANT angular
acceleration, there are set kinematics
equations.
Linear Kinematics
Rotational Kinematics
Circular Motion
September 29, 2016
o
Be able to recognize when a problem is in UNIFORM circular motion or in NON­UNIFORM circular motion
Be able to use kinematics equations to solve problems
§
An electric fan goes from rest to 1800 rpm
in 4 s. What is its angular acceleration?
o
Be able to recognize when a problem is in UNIFORM circular motion or in NON­UNIFORM circular motion
Be able to use kinematics equations to solve problems
§
A turntable starts at rest and has angular
acceleration, α. If the angular speed at the
end of 1 rotation is ω, what is the angular
speed (in terms of ω) after only 1/4 of a
rotation?
Circular Motion
September 29, 2016
Understand the difference between tangential components and radial components and be able to use equations the relate §
the two
Tangential acceleration and centripetal acceleration
Suppose we have an object moving
nonuniformly in a circle. At some point, the
acceleration vector is directed as shown:
v
a
How would we draw tangential acceleration?
How would we draw centripetal acceleration?
How are these 2 vectors related to the actual acceleration?
o
Be able to relate these words to circular motion, know their equations and their units
period, arc length, angular position, tangential velocity, angular velocity, centripetal acceleration, angular acceleration, §
tangential acceleration
Relating more equations:
at = dvt
dt
vt = ωr at = αr
Circular Motion
September 29, 2016
o
Be able to relate these words to circular motion, know their equations and their units
period, arc length, angular position, tangential velocity, angular velocity, centripetal acceleration, angular acceleration, §
tangential acceleration
Astronauts use a centrifuge to simulate the acceleration of a rocket launch. The centrifuge take 30 s to speed up from rest to its top speed of 1 rotation every 1.3 s. The astronaut is strapped into a seat 6.0 m from the axis. What is the astronaut's tangential acceleration during the first 30 s?
How many g's of acceleration does the astronaut experience when the device is rotating at top speed? (Each 9.8 m/s2 of acceleration is 1 g.
ð
Know the properties of Circular Motion o
Be able to convert radians to degrees to revolutions
Understand the difference between tangential components and radial components and be able to use equations §
the relate the two
o
Be able to relate these words to circular motion, know their equations and their units
period, arc length, angular position, tangential velocity, angular velocity, centripetal acceleration, angular §
acceleration, tangential acceleration
o
Be able to recognize when a problem is in UNIFORM circular motion or in NON­UNIFORM circular motion
Be able to use kinematics equations to solve problems §
Be able to use calculus to solve problems with a changing angular acceleration
§
o
Be able to graph angular position vs. time, angular velocity vs. time, and angular acceleration vs. time for a given scenario (or describe the motion of given graphs)
Homework time!