1. Rotational Kinematics and Dynamics
Test: Friday 9/21

2. Circular Motion
For a particle in uniform circular motion, the velocity vector v remains constant in magnitude, but it continuously changes its direction.

3. A Particle in Uniform Circular Motion

4. Angular Position: θ

5. Angular Position q

6. Angular Position q
Arc length s, measured in radians:

7. Angular Velocity w

8. Sign of 

9. Connections Between Linear & Rotational Quantities

10. Angular Acceleration a

11. Comparison to 1-D Kinematics
Angular Linear
By convention, , ,  are positive if they are in the counterclockwise direction.

12. Connections Between Linear & Rotational Quantities
And for a point at a distance (r) from the rotation axis:
x = rv = r a = r

13. More Connections Between Linear & Rotational Quantities

14. More Connections Between Linear & Rotational Quantities
This merry-go-round has both tangential and centripetal acceleration.
Speeding up

15. Example Problem #1
As the wind dies, a windmill that had been rotating at w = 2.1 rad/s begins to slow down at a constant angular acceleration of a = rad/s2.
How long does it take for the windmill to come to a complete stop?

16. Concept Question (a) w>0 and a>0 (b) w>0 and a<0
The fan blade shown is slowing down. Which option describes a and w?
(a) w>0 and a>0
(b) w>0 and a<0
(c) w<0 and a>0
(d) w<0 and a<0

17. Example Problem #2
To throw a curve ball, a pitcher gives the ball an initial angular speed of rad/s. When the catcher gloves the ball s later, its angular speed has decreased (due to air resistance) to rad/s.
(a) What is the ball’s angular acceleration, assuming it to be constant?
(b) How many revolutions does the ball make before being caught?

18. Example Problem #3
A car’s tachometer indicates the angular velocity w of the crank shaft in rpm. A car stopped at a traffic light has its engine idling at 500 rpm. When the light turns green, the crankshaft’s angular velocity speeds up at a constant rate to 2500 rpm in a time interval of 3.0 s.
How many revolutions does the crankshaft make in this time interval?

19. The next few slides have more example problems if you would like more practice on your own.
The answers will be posted on the website tomorrow

20. Example Problem #4
On a certain game show, contestants spin the wheel when it is their turn. One contestant gives the wheel an initial angular speed of 3.40 rad/s. It then rotates through 1.25 revolutions and comes to rest on BANKRUPT.
Find the wheel’s angular acceleration, assuming it to be constant.
(b) How long does it take for the wheel to come to rest?

21. Example Problem #5
A pulley rotating in the counterclockwise direction is attached to a mass suspended from a string. The mass causes the pulley’s angular velocity to decrease with a constant angular acceleration a = rad/s2.
(a) If the pulley’s initial angular velocity is w0 = 5.40 rad/s, how long does it take for the pulley to come to rest?
Through what angle does the pulley turn during this time?
If the radius of the pulley is 5.0 cm, through what distance is the mass lifted?

22. Example Problem #6
CDs turn with a variable w that keeps the tangential speed vt constant.
Find the angular speed w that a CD must have in order to give it a linear speed vt = 1.25 m/s when the laser beam shines on the disk
(a) at 2.50 cm from its center (b) at 6.00 cm from its center

23. Example Problem #7
Suppose the centrifuge is just starting up, and that it has an angular speed of 8.00 rad/s and an angular acceleration of 95.0 rad/s2.
(a) What is the magnitude of the centripetal, tangential, and total acceleration of the bottom of a tube?
(b) What angle does the total acceleration make with the direction of motion?

24. Board Meeting Expectations:
Torque and Rotational Acceleration Inquiry Lab This is an RA grade and you will have a board meeting on Fri. 9/14
Goal: To explore the relationship between torque and angular acceleration
Board Meeting Expectations:
Data Table and Observations
If you average any data to use in your calculations you must show the average and the standard deviation value for it
Calculations (1 of each type shown)
Graph w/an equation for your line of best fit
Explain what the slope represents
Explain the relationship between torque and angular acceleration

25. Rotational Inertia
Is a measure of an object's resistance to changes to its rotation.
It is the inertia of a rotating body with respect to its rotation and measured in kg·m²
It is the rotational counterpart to linear mass
(Will also be called the moment of inertia)

26. Let’s recall Newton’s 2nd Law (Linear Motion)

27. Newton's Second Law for Rotation

28. Rotational Inertia

29. Example Problem #8
Rank the three masses according to their rotational inertia about that axis from greatest to least A B C

30. Example Problem #9
Find the moment of inertia of the system below about point O.

31. Example Problem #10
Three beads, each of mass m, are arranged along a rod of negligible mass and length L. Find the rotational inertia when the axis of rotation is through
the center bead
one of the beads on the end.

32. Rotational Inertia for Common Shapes

33. Parallel Axis Theorem
What if an object is rotating about an axis parallel to an axis through the center of mass and a distance d from it?

34. Parallel Axis Theorem

35. Using the Parallel Axis Theorem instead…. Example #10 again
Three beads, each of mass m, are arranged along a rod of negligible mass and length L. Find the rotational inertia when the axis of rotation is through one of the beads on the end.

36. APP1 vs APPC
PULLEYS!!

37. Let’s just ignore the pulley!! We can’t just ignore the pulley!!
APP1 vs APPC
Let’s just ignore the pulley!!
We can’t just ignore the pulley!!
Two unequal masses, m1 and m2, are attached to a string that passes over a massless pulley. Determine the linear acceleration of the masses in terms of m1, m2, and g. 
Two unequal masses, m1 and m2, are attached to a string that passes over a pulley with a mass mp and radius r.  Determine the linear acceleration of the masses in terms of m1, m2, mp, g, and r.