1. Finish Momentum Start Spinning Around
March 22, 2006

2. Watsup? Quiz on Friday Watch for still another WebAssign
Last part of energy conservation through today.
Watch for still another WebAssign
Full calendar for the remainder of the semester is on the website near the end of the last set of PowerPoint slides.
Next exam is next week!

3. Last Time

4. We therefore define the center of mass as

5. Center of Mass, Coordinates
The coordinates of the center of mass are
where M is the total mass of the system

6. Center of Mass, position
The center of mass can be located by its position vector, rCM
ri is the position of the i th particle, defined by

7. Center of Mass, Example Both masses are on the x-axis
The center of mass is on the x-axis
The center of mass is closer to the particle with the larger mass

8. Center of Mass, Extended Object
Think of the extended object as a system containing a large number of particles
The particle distribution is small, so the mass can be considered a continuous mass distribution

9. Center of Mass, Extended Object, Coordinates
The coordinates of the center of mass of a uniform object are

10. Center of Mass, Extended Object, Position
The position of the center of mass can also be found by:
The center of mass of any symmetrical object lies on an axis of symmetry and on any plane of symmetry

11. Density r
Usually, r is a constant but NOT ALWAYS!

12. Center of Mass, Example
An extended object can be considered a distribution of small mass elements, Dm=rdxdydz
The center of mass is located at position rCM

13. For a flat sheet, we define a mass per unit area … s

14. Linear Density m

15. Center of Mass, Rod Let's just do it!
Find the center of mass of a rod of mass M and length L
The location is on the x-axis (or
yCM = zCM = 0)
xCM = L / 2
Let's just do it!

16. A golf club consists of a shaft connected to a club head
A golf club consists of a shaft connected to a club head. The golf club can be modeled as a uniform rod of length L and mass m1 extending radially from the surface of a sphere of radius R and mass m2. Find the location of the club’s center of mass, measured from the center of the club head. R m2 L m1

17. Motion of a System of Particles
Assume the total mass, M, of the system remains constant
We can describe the motion of the system in terms of the velocity and acceleration of the center of mass of the system
We can also describe the momentum of the system and Newton’s Second Law for the system

18. Velocity and Momentum of a System of Particles
The velocity of the center of mass of a system of particles is
The momentum can be expressed as
The total linear momentum of the system equals the total mass multiplied by the velocity of the center of mass

19. Acceleration of the Center of Mass
The acceleration of the center of mass can be found by differentiating the velocity with respect to time

20. Newton’s Second Law for a System of Particles
Since the only forces are external, the net external force equals the total mass of the system multiplied by the acceleration of the center of mass:
SFext = M aCM
The center of mass of a system of particles of combined mass M moves like an equivalent particle of mass M would move under the influence of the net external force on the system

21. Momentum of a System of Particles
The total linear momentum of a system of particles is conserved if no net external force is acting on the system
MvCM = ptot = constant when SFext = 0

22. Motion of the Center of Mass, Example
A projectile is fired into the air and suddenly explodes
With no explosion, the projectile would follow the dotted line
After the explosion, the center of mass of the fragments still follows the dotted line, the same parabolic path the projectile would have followed with no explosion

23. Rotation of a Rigid Object about a Fixed Axis
Chapter 10
Rotation of a Rigid Object
about a Fixed Axis

24. Rigid Object A rigid object is one that is nondeformable
The relative locations of all particles making up the object remain constant
All real objects are deformable to some extent, but the rigid object model is very useful in many situations where the deformation is negligible

25. Angular Position Axis of rotation is the center of the disc
Choose a fixed reference line
Point P is at a fixed distance r from the origin

26. Angular Position, 2
Point P will rotate about the origin in a circle of radius r
Every particle on the disc undergoes circular motion about the origin, O
Polar coordinates are convenient to use to represent the position of P (or any other point)
P is located at (r, q) where r is the distance from the origin to P and q is the measured counterclockwise from the reference line

27. Angular Position, 3
As the particle moves, the only coordinate that changes is q
As the particle moves through q, it moves though an arc length s.
The arc length and r are related:
s = q r

28. Radian This can also be expressed as
q is a pure number, but commonly is given the artificial unit, radian
One radian is the angle subtended by an arc length equal to the radius of the arc

29. Conversions Comparing degrees and radians
Converting from degrees to radians
θ [rad] = [degrees]

30. Angular Position, final
We can associate the angle q with the entire rigid object as well as with an individual particle
Remember every particle on the object rotates through the same angle
The angular position of the rigid object is the angle q between the reference line on the object and the fixed reference line in space
The fixed reference line in space is often the x-axis

31. Angular Displacement
The angular displacement is defined as the angle the object rotates through during some time interval
This is the angle that the reference line of length r sweeps out

32. Average Angular Speed
The average angular speed, ω, of a rotating rigid object is the ratio of the angular displacement to the time interval

33. Instantaneous Angular Speed
The instantaneous angular speed is defined as the limit of the average speed as the time interval approaches zero

34. Angular Speed, final Units of angular speed are radians/sec
rad/s or s-1 since radians have no dimensions
Angular speed will be positive if θ is increasing (counterclockwise)
Angular speed will be negative if θ is decreasing (clockwise)

35. Average Angular Acceleration
The average angular acceleration, a,
of an object is defined as the ratio of the change in the angular speed to the time it takes for the object to undergo the change:

36. Instantaneous Angular Acceleration
The instantaneous angular acceleration is defined as the limit of the average angular acceleration as the time goes to 0

37. Angular Acceleration
Units of angular acceleration are rad/s² or s-2 since radians have no dimensions
Angular acceleration will be positive if an object rotating counterclockwise is speeding up
Angular acceleration will also be positive if an object rotating clockwise is slowing down

38. Angular Motion, General Notes
When a rigid object rotates about a fixed axis in a given time interval, every portion on the object rotates through the same angle in a given time interval and has the same angular speed and the same angular acceleration
So q, w, a all characterize the motion of the entire rigid object as well as the individual particles in the object

39. Directions Using this convention, w is a VECTOR!
Strictly speaking, the speed and acceleration (w, a) are the magnitudes of the velocity and acceleration vectors
The directions are actually given by the right-hand rule
Using this convention, w is a VECTOR!

40. Hints for Problem-Solving
Similar to the techniques used in linear motion problems
With constant angular acceleration, the techniques are much like those with constant linear acceleration
There are some differences to keep in mind
For rotational motion, define a rotational axis
The choice is arbitrary
Once you make the choice, it must be maintained
The object keeps returning to its original orientation, so you can find the number of revolutions made by the body

41. Rotational Kinematics
Under constant angular acceleration, we can describe the motion of the rigid object using a set of kinematic equations
These are similar to the kinematic equations for linear motion
The rotational equations have the same mathematical form as the linear equations

42. Rotational Kinematic Equations

43. The derivations are similar to what we did with kinematics. Example:

44. Comparison Between Rotational and Linear Equations

45. Relationship Between Angular and Linear Quantities
Displacements
Speeds
Accelerations
Every point on the rotating object has the same angular motion
Every point on the rotating object does not have the same linear motion

46. A dentist's drill starts from rest. After 3
A dentist's drill starts from rest. After 3.20 s of constant angular acceleration, it turns at a rate of 2.51  104 rev/min. (a) Find the drill's angular acceleration. (b) Determine the angle (in radians) through which the drill rotates during this period.

47. An airliner arrives at the terminal, and the engines are shut off
An airliner arrives at the terminal, and the engines are shut off. The rotor of one of the engines has an initial clockwise angular speed of rad/s. The engine's rotation slows with an angular acceleration of magnitude 80.0 rad/s2. (a) Determine the angular speed after 10.0 s. (b) How long does it take the rotor to come to rest?

48. A centrifuge in a medical laboratory rotates at an angular speed of rev/min. When switched off, it rotates 50.0 times before coming to rest. Find the constant angular acceleration of the centrifuge.

49. Speed Comparison
The linear velocity is always tangent to the circular path
called the tangential velocity
The magnitude is defined by the tangential speed

50. Acceleration Comparison
The tangential acceleration is the derivative of the tangential velocity

51. Speed and Acceleration Note
All points on the rigid object will have the same angular speed, but not the same tangential speed
All points on the rigid object will have the same angular acceleration, but not the same tangential acceleration
The tangential quantities depend on r, and r is not the same for all points on the object

52. Centripetal Acceleration
An object traveling in a circle, even though it moves with a constant speed, will have an acceleration
Therefore, each point on a rotating rigid object will experience a centripetal acceleration

53. Resultant Acceleration
The tangential component of the acceleration is due to changing speed
The centripetal component of the acceleration is due to changing direction

54. Rotational Motion Example
For a compact disc player to read a CD, the angular speed must vary to keep the tangential speed constant (vt = wr)
At the inner sections, the angular speed is faster than at the outer sections