1. 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

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 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

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 = w o + αt θ = θ o + w o t + 0.5 αt 2 a = α * r

7. Example time You accelerate a bicycle wheel from rest for 4.4 seconds at an angular acceleration of 3.3 rad/sec 2. 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 * v 2 / r or, a = v 2 / 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 don’t 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 / r 2 = M1 v 2 / r Solving this for v you get: v 2 = 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 4*10 8 m. What is the orbital velocity of the moon around the earth. Mass of the earth is 6 * 10 24 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: P 2 = [4 pi*pi / G M2] * r 3 Your example. Mass of the earth is 6 * 10 24 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