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 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 = wo + αt • θ = θo + wot + 0.5 αt2 • 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/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 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 / 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 4*108 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 pi*pi / G M2] * r3 • 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