Goal: To understand angular motionsObjectives:

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 circle2) degrees3) 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-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?

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