Goal: To understand angular motions

1
Goal To understand angular motions

• Objectives
• To learn about angles
• To learn about angular velocity
• To learn about angular acceleration
• To learn about centrifugal force
• To explore planetary orbits
• Note this lecture is designed to go for 2 class
  periods and will be the only chapter 5 lecture

2
Circular Motion

• Previously we examined speed and velocity.
• However these were movements in a straight line.
• Sometimes motions are not straight, but circular.

3
Angle

• Instead of moving a distance X we can rotate an
  angular distance ?
• So, ? is the angular equivalent to X
• Furthermore X ? 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

4
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)

5
Lets do an example.

• You are 0.5 m from the center of a
  merry-go-round.
• 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?

6
Angular acceleration

• The linear equations once again transform right
  to the linear
• w wo at
• ? ?o wot 0.5 at2
• a a r

7
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?

8
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.

9
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

10
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 dont need the mass of the roller coaster
  here.

11
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

12
Orbit example

• The moon orbits the earth at a distance of 4108
  m.
• What is the orbital velocity of the moon around
  the earth.
• Mass of the earth is 6 1024 kg

13
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 pipi / 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

14
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