Transcript Lecture 07 - WebPhysics

Goal: To understand angular
motions
Objectives:
1) To learn about angles
2) To learn about angular velocity
3) To learn about angular acceleration
4) To learn about centrifugal force
5) To explore planetary orbits
Note this lecture is designed to go for 2 class
periods and will be the only chapter 5 lecture
Circular Motion
• Previously we examined speed and
velocity.
• However these were movements in a
straight line.
• Sometimes motions are not straight, but
circular.
Angle
• Instead of moving a distance X we can rotate an
angular distance θ
• So, θ is the angular equivalent to X
• Furthermore X = θ * r where r is the radius of the
circle you are rotating on
• Units for angle:
1) radians (most used). There are 2 pi radians
in a circle
2) degrees
3) revolutions – one circle is one revolution
Around and around
• If you rotate in a circle there will be a rate you
rotate at.
• That is, you will move some angle every second.
• w = angular velocity = change in angle / time
• Units of w are radians/second or
degrees/second
• If you want a linear speed, the conversion is:
• V = radius * angular velocity (in radians /
second)
Lets do an example.
• You are 0.5 m from the center of a merry-goround.
• If you go around the merry-go-round once every
3.6 seconds (hint, how many degrees in a circle)
then what is your angular velocity in
degrees/second.
• There are 2 pi radians per circle.
• A) What is your angular velocity in radians per
second?
• B) What is your linear velocity in meters per
second?
Angular acceleration
• The linear equations once again transform
right to the linear
• w = wo + αt
• θ = θo + wot + 0.5 αt2
• a=α*r
Example time
• You accelerate a bicycle wheel from rest for 4.4
seconds at an angular acceleration of 3.3 rad/sec2.
The radius of the wheel is 0.72 meters.
• A) What will the angular velocity of the bicycle
wheel be after the 4.1 seconds?
• B) If the bicycle was moving what would its linear
velocity be after the 4.1 seconds?
• C) How far (in angle) will the bicycle have rotated
in 4.1 seconds?
• D) How far in meters would the bicycle have
traveled in 4.1 seconds?
Centripetal vs Centrifugal force
• These two are very similar.
• Centripetal force is a force that pulls you to
the center.
• Gravity is an example here.
• When you are in circular motion,
centrifugal force will try to push you out,
and try to cancel out the centripetal force.
Equation
• Centrifugal force: F = m * v2 / r
• or, a = v2 / r
•
•
•
•
Example time:
A 500 kg car goes around a 50 m turn.
The frictional coefficient is 0.2
What is the maximum velocity the car can go
without crashing (that is to say that the car does
not slide in the turn)? This problem takes 2 steps
Another example
• A roller coaster does a loop de loop.
• If the radius of the loop-de-loop is 25
meters find the minimum velocity the
coaster must have in order to stay on the
tracks
• Hint, think about what the outwards
acceleration at the top of the loop will need
to be.
• No, you don’t need the mass of the roller
coaster here.
Orbits
• This leads to orbits.
• In a circular orbit (where M1 is orbiting M2)
the gravitational force is canceled by the
centrifugal force.
• That is to say that G M1 M2 / r2 = M1 v2 / r
• Solving this for v you get:
• v2 = G M2 / r
this is the orbital velocity
• NOTE: r is the distance to the center not
the surface
Orbit example
• The moon orbits the earth at a distance of
4*108 m.
• What is the orbital velocity of the moon
around the earth.
• Mass of the earth is 6 * 1024 kg
Orbital period
• If you take that the circumference of the orbit is
2pi r combined with the orbital velocity you will
find that the time it takes to do a full orbit around
M2 is:
• P2 = [4 pi*pi / G M2] * r3
• Your example.
• Mass of the earth is 6 * 1024 kg
• Find the distance at which the orbital period
around the Earth is 1 day (86400 s) – note this is
called Geosynchronous
Conclusion
• We have learned about the parallels
between linear motions and angular
motions
• We have learned about how to use
centrifugal force
• We have learned about orbits