1. Rotational Motion and the Law of Gravity
7.1 Measuring Rotational Motion

2. When something spins around a certain point its called – circular motion
A circle of radius “r” has a circumference 2πr and has an angular measure of 360°
Circular motion is describes in terms of the angle that the object moves.
when the CD (compact disc) in Figure 1 rotates about its center—each point in the object follows a circular arc. Consider a line from the center of the CD to its edge. Each pit used to record sound along this line moves through the same angle in the same amount of time. The rotation angle is the amount of rotation and is analogous to linear distance. We define the rotation angle Δθ to be the ratio of the arc length to the radius of curvature:Δθ=Δsr.

3. ∆ϴ = ∆s r Angles in physics are measured in radians (rad)
The angular displacement in radians (∆ϴ) is the ratio of the change in arc length (s) to the radius (r) of a circle
∆ϴ = ∆s r
Formula:
Units cancel out and rad is used
Conversions:
2π radians = ° = 1 rev
And 1 rad = 1800/ π = 57.30

5. Angular Speed describes the rate of rotation Formula: ω = ∆ϴ / ∆ t
Units: rad / sec
Greek letter omega (ω) = angular speed
Average angular speed = angular displacement / time interval
When an object spins, we can describe how fast its going in terms of either:
degrees/sec or revolutions/sec or radians/sec
Example Problem:
Earth makes one rotation (3600) about its axis
in one day (24 hours)
If = 2 π rad and hours = sec
And 2 x π = 6.28
Then,
2 π rad / sec =
Earth’s angular speed = 7.27 x rad/sec

6. In a certain game show, contestants spin a wheel when it is their turn
In a certain game show, contestants spin a wheel when it is their turn. One contestant gives the wheel an initial angular speed of 3.20 rad/s. It then rotates through one-and-one-quarter revolutions and comes to rest on the BANKRUPT space. Through what angle has the wheel turned when its angular speed is 1.90 rad/s?

7. Average angular acceleration = change in angular speed / time interval
Angular acceleration occurs when angular speed changes
Formula α = ∆ω
∆ t
Greek letter α = alpha
Average angular acceleration = change in angular speed / time interval

8. Comparing angular and linear quantitiesx ϴ v ω a α

9. 7.2 Tangential and Centripetal Acceleration
Objects in circular motion have a tangential speed
Any two objects have the same angular speed and angular acceleration regardless of distance from center but….
If the two objects are different distances from the axis of rotation, they have different tangential speeds
(instantaneous linear speed at that point)

10. Tangential Speed Vtan = r x ω
Formula: distance from the axis (r) x instantaneous angular speed (ω)
Vtan = r x ω
tangential speed and angular speed

11. atan = r x α Tangential Acceleration
instantaneous linear acceleration is tangent to the circular path
atan = r x α
tangential acceleration = distance from the axis x angular acceleration

12. Centripetal Acceleration
Centripetal - “towards the center”
Since Velocity is a vector there are 2 ways an acceleration can be produced:
change in magnitude
and/or change in direction
For a car moving in a circular path with constant speed the object is accelerating due to a change in direction.
Experienced by any object that travels in a curved path
ac = vtan2 r
centripetal acceleration = (tangential speed)2
distance from the axis
ac = r ● ω2

13. Since Vtan = r ωavg
And ac = Vtan2 r
Sooo, ac = (r2 ● ω2)
Or ac = r ● ω2

14. 7-3 Causes of Circular Motion
Force that maintains circular motion
Centripetal Force - any force towards center
Examples: Earth’s gravitational pull on moon or the electric force that pulls electrons around atomic nuclei
According to Newton’s 2nd Law:
Fc = m x ac or
Fc = m ● vtan2 or r
Fc = m x r x ω2
Units are in Newtons (N)
Centripetal force

15. 3 Types of Acceleration
Linear (tangential) – rate of change of speed… due to change in speed
units: m/s2
2) Angular – rate of change of angular speed – due to change in speed
units: rad/s2
3) Centripetal – a center-seeking acceleration – due to change in direction

16. Newton’s Law of Universal Gravitation
Why do our planets stay in the sun’s orbit?
Why does the moon stay in orbit around the Earth?
Mutual force of attraction between 2 objects
According to Newton’s 2nd Law of Motion
Fg = G m1m2 r2
G = constant of universal gravitation = x N m2
kg2

19. Launch Speed less than 8000 m/s
Projectile falls to Earth
Launch Speed less than 8000 m/s
Projectile falls to Earth
Launch Speed equal to 8000 m/s
Projectile orbits Earth - Circular Path
Launch Speed greater than 8000 m/s
Projectile orbits Earth - Elliptical Path