1. Rotation about a fixed axis
Rotational Motion
Rotation about a fixed axis

2. Rotational Motion
Translation
Rotation
Rolling

3. Describing Angular Motion
Angular Kinematics
Describing Angular Motion

4. Angular Position - q Displacement is now defined as angle of rotation.
Polar coordinates (r, q) will make it easier to notate this motion. Dq

5. Dq = q1 – q0 Angular Position - q
For a circle: 2p=circumference/radius
So: q = s/r or s = qr
s => arc length
r => radius
q => angle, measured in radians
Remember: 360º = 2p radians
Displacement:
Dq = q1 – q0

6. Angular Velocity - w Average angular velocity
Instantaneous angular velocity

7. Angular Direction Right-hand coordinate system
w is positive when q is increasing, which happens in the counter-clockwise direction
w is negative when q is decreasing, which happens in the clockwise direction

8. Angular Acceleration - a
Average Angular Acceleration
Instantaneous Angular Acceleration

9. At what time(s) does w = 0 rad/s?
example 1
q = (t3 - 27t + 4)rad
w = ?
a = ?
At what time(s) does w = 0 rad/s?

10. At t = 0s: w = 5 rad/s and q = 2 rad.
example 2
a = (5t3 – 4t)rad/s2
At t = 0s: w = 5 rad/s and q = 2 rad.
w = ?
q = ?

11. Linear to Angular Conversions
Position: s = qr
Velocity: v = wr
Acceleration: a = ar
Remember, also, centripetal accleration: ac = v2/r =w2r

12. Angular Kinematics *constant a

13. example 3
A turntable starts from rest and begins rotating in a clockwise direction. 10 seconds later, it is rotating at 33.3 revolutions per minute.
What is the final angular velocity (rad/s)?
What is the average angular acceleration?
How far did a point 10 cm from the center travel in that 10 seconds, both angularly and linearly?

14. example 4 In an astronaut training centrifuge (r = 15m):
What constant w would give 11g’s?
How fast is this in terms of linear speed?
What is the translational acceleration to get to this speed from rest in 2 minutes?

15. Moment of Inertia
“Rotational Mass”

16. Moment of Inertia Description of the distribution of mass
Measure of an object’s ability to resist a change in rotation
Note: r is the perpendicular distance from the particle to the axis of rotation.

17. example
Two children (m=40 kg) are on the teacup ride at King’s Dominion. The childrens’ center of mass is 0.75 m from the middle of the cup. What is the moment of inertia for children in the teacup?

18. Moment of Inertia for Continuous Mass Distribution

19. Calculate I for a hoop
A uniform hoop: mass (M); radius (R)

20. Calculate I for a thin rod
A uniform thin rod: mass (M); length (L); rotating about its center of mass.

21. Common Moments of Inertia

22. Parallel Axis Theorem
To calculate the moment of inertia around any axis….
Where d is the distance between the center of mass and the axis of rotation

23. Calculate I for a thin rod rotating around the end
A uniform thin rod: mass (M); length (L); rotating about one end.

24. The kinetic energy an object has due to its rotational velocity.
Rotational Energy
The kinetic energy an object has due to its rotational velocity.

25. Rotational Kinetic Energy

26. example
If you roll a disk and a hoop (of the same mass) down a ramp, which will win?

27. The tendency of a force to cause angular motion
Torque
The tendency of a force to cause angular motion

28. Torque
Torque is dependent on the amount and location of the force applied to an object.
Where r is the distance between the pivot point and the force and q is the angle between r and F.

29. example 5
A one piece cylinder has a core section that protrudes from a larger drum. A rope wrapped around the large drum of radius, R, exerts a force, F1, to the right, while a rope wrapped around the core, radius r, exerts a force, F2 downward..
Calculate the net torque, in variables.
If F1=5 N, R = 1 m, F2=6 N, and r = 0.5 m, calculate the net torque.

30. The “other” vector multiplication
Cross Product
The “other” vector multiplication

31. Cross product Results in a vector quantity
Calculates the perpendicular product of two vectors
The product of any two parallel vectors will always be zero.

32. The Right Hand Rule
Rotational Direction is defined by the axis that it rotates around.
Point your fingers along the radius
Curl them in the direction of the force
Your thumb will be pointing in the direction of the rotation.

33. Cross Product

34. Calculating Cross Product

35. Finding the Determinant

36. example 6 Find the cross product of 7i + 5j – 3k and 2i-8j + 7k.
What is the angle between these two vectors?

37. example 7
A plumber slips a piece of scrap pipe over his wrench handle to help loosing a pipe fitting. He then applies his full weight (900 N) to the end of the pipe by standing on it. The distance from the fitting to his foot is 0.8 m, and the wrench and pipe make a 19º angle with the ground. Find the magnitude and direction of the torque being applied.

38. example 7 continued…
r = 0.8m
q = 19º
F = 900 N

39. example 8
One force acting on a machine part is F = (-5i + 4j)N. The vector from the origin to the point applied is r = (-0.5i + 0.2j)m.
Sketch r and F with respect to the origin
Determine the direction of the force with the right hand rule.
Calculate the torque produced by this force.
Verify that your direction agrees with your calculation.

40. Equilibrium

41. Conditions for Equilibrium
Net force in all directions equals zero
Net torque about any point equals zero

42. example 9
A 2 kg seesaw has one child (30 kg) sitting 2.5 from the pivot. At what distance from the pivot should a 25 kg child sit to balance the seesaw?

43. example 10
Find Tcable.

44. example 11
Find T1 and T2.

45. example 12
Find m.

46. When the sum of the torques does not equal zero
Newton’s Second Law
When the sum of the torques does not equal zero

47. Newton’s Second Law

48. example 13
a = ?
Fpivot L
Fnormal
Fgravity

49. example 14 FT Fg
Pulley – M1, R Block – M2 Find a and FT

50. Angular Momentum

51. For a particle
Remember linear momentum:
So angular momentum:

52. Angular momentum and Torque

53. For a system…

54. To check…

55. Conservation
If net torque is zero, then the angular momentum is constant.

56. Conservation
For rotation about a fixed axis, we can say:

57. example
A star with a radius of 10,000 km rotates about its axis with a period of 30 days. If it undergoes a supernova explosion and collapses into a neutron star with a radius of 3 km, what is its new period?