1. Show how the rotational kinematics equation are just “Greek” versions of the linear kinematics equations …

2. What do you notice? 𝑣 = 𝑣 𝑜 +𝑎𝑡 𝑥 = 𝑥 𝑜 + 𝑣 𝑜 𝑡+ 1 2 𝑎 𝑡 2
Equations for kinematics assuming
constant acceleration
Eq 1
𝑣 = 𝑣 𝑜 +𝑎𝑡
(no x)
𝑥 = 𝑥 𝑜 + 𝑣 𝑜 𝑡 𝑎 𝑡 2
(no V)
Eq 2
What do you notice?
𝑥 = 𝑥 𝑜 + 𝑣+ 𝑣 𝑜 2 𝑡
Eq 3
(no a)
𝑣 = 𝑣 𝑜 2 +2a( 𝑥 − 𝑥 𝑜 )
(no t)
Eq 4
𝑥 = 𝑥 𝑜 + 𝑣 𝑡− 𝑎 𝑡 2
Extra
(no Vo)

3. Use them in exactly the same way!
Now:
When angular acceleration () is constant……
Recall:
When acceleration (a) is constant……
vf = vi + at
x = vit + ½ at2
vf2 = vi2 + 2ax
x = ½ (vi+vf)t
f = i + t
 = it + ½ t2
f2 = i2 + 2
 = ½ (i + f)t
Notice that these four equations are exactly the same as the previous four, with these substitutions:
x 
v 
a 
Use them in exactly the same way!

4. Show sensible rounding…
A Centrifuge can accelerate from rest at a constant angular acceleration of 7.0 rad s-2, taking 3 minutes to reach top speed
What is the final angular velocity?
How many rotations does it turn through during this time?
Show sensible rounding…

5. Answers: 𝜔 𝑓 = 𝜔 𝑖 +α𝑡 𝜔 𝑓 =0+7.0×180 𝜔 𝑓 =1300 𝑟𝑎𝑑 𝑠 −1
𝜔 𝑓 =1300 𝑟𝑎𝑑 𝑠 −1
Θ= 𝜔 𝑖 𝛼 𝑡 2
Θ= × 180 2
Θ= rad
# of turns = /2 = rev

6. What is the angular acceleration of a compact disc that begins at rest and accelerates to 36 rads-1 in 1.5 s?
ω0 = 0
ω = 36 rads-1
α = ?
t = 1.5s
θ = ignore
Audio CD players read their discs at a constant 150 kB/s and thus must vary the disc's rotational speed from around 500 rpm (~ 8 Hz), when reading at the innermost edge, to 200 rpm (~3.5 Hz) at the outer edge.

7. A tire on car has an angular acceleration of –5 rad/s2 applied to it
A tire on car has an angular acceleration of –5 rad/s2 applied to it. If the wheel was originally turning at 80 rad/s and turned through 500 rad during the deceleration, what was the final speed of the wheel?
θ = 500 rad
ω0 = 80 rad/s
ω = ?
α = –5 rad/s2
t = ignore
ω2 = ω02+ 2αθ
ω= 37 rad/s

8. Rotational Graphs…

10. 1) vs t Open your laptop and answer the next 14 slides:
Which is the correct description
of the motion needed to make this
angular velocity vs time graph?
Spinning at a constant speed in + direction
Spinning at a constant speed first in the – direction and then in the + direction
Speeding up in the – direction and then turning around and speeding up in + direction
Speeding up in the + direction the whole time
Slowing down in the – direction and then turning around and speeding up in + direction
Velocity
time

11. 2) Which difference in the rotational motion of the
purple top compared to the blue top
as shown on this angular velocity vs time graph?
The purple top is spinning faster than the blue top and they are rotating in opposite directions
The purple top is spinning faster than the blue top and they are rotating in the same direction
The purple top is slowing down faster than the blue top is speeding up and they are going in opposite directions
The purple top is slowing down faster than the blue top is speeding up and they are going in the same direction
time
Velocity

12. 3) θ vs t  ω vs t
Position
Given this angular position vs time graph (red), which angular velocity vs time graph (blue) depicts the same motion? A. B. C. D. E.
time
Velocity
Velocity
time
time
time
time
time
Velocity
Velocity
Velocity

13. 4) θ vs t  ω vs t
Given this angular velocity vs time graph (blue), which angular position vs time graph (red) depicts the same motion? A. B. C. D. E.
Velocity
time
Position
Position
time
time
Position
Position
Position
time
time
time

14. ω vs t  θ vs t
Given this angular velocity vs time graph (blue), which angular position vs time graph (red) depicts the same motion? A. B. C. D. E.
Velocity
time
Position
Position
time
time
Position
Position
Position
time
time
time

15. 5) vs t  θ vs t
Given this angular velocity vs time graph (blue), which angular position vs time graph (red) depicts the same motion? A. B. C. D. E.
Velocity
time
Position
Position
time
time
Position
Position
Position
time
time
time

16. 6) ω vs t A) ¾ B) 12 C) 24 D) 48 E) None of These F) Cannot Determine
Given the angular velocity versus time graph to the right, what is the total angular displacement of the object?
A) ¾
B) 12
C) 24
D) 48
E) None of These
F) Cannot Determine 8 6 4 2
Velocity (m/s)
t (s) 2 4 6 8 10 12 14 16 -2 -4 -6 -8

17. 7) ω vs t A) 0 rad B) 18 rad C) 36 rad D) None of These
Given the angular velocity versus time graph to the right, what is the total angular displacement of the object?
A) 0 rad
B) 18 rad
C) 36 rad
D) None of These
E) Cannot Determine 8 6 4 2
Velocity (m/s)
t (s) 2 4 6 8 10 12 14 16 -2 -4 -6 -8

18. 8) ω vs t  α vs t
Velocity
Given this angular velocity vs time graph (blue), which angular acceleration vs time graph(green) depicts the same motion? A. B. C. D. E.
time
acceleration
acceleration
time
time
acceleration
acceleration
time
time
acceleration
time

19. 9) ω vs t  α vs t
Velocity
time
Given this angular velocity vs time graph (blue), which angular acceleration vs time graph(green) depicts the same motion? A. B. C. D. E.
acceleration
acceleration
time
time
acceleration
acceleration
time
acceleration
time
time

20. 10) θ vs t  α vs t
Given this angular position vs time graph (red), what is the direction of angular acceleration? + –
No direction
Not enough information
Position
time

21. 11) θ vs t  α vs t
Given this angular position vs time graph (red), what is the direction of angular acceleration? + –
No direction
Not enough information
Position
time

22. 12) ω vs t  α vs t
Velocity
time
Given this angular velocity vs time graph (blue), which angular acceleration vs time graph(green) depicts the same motion? A. B. C. D. E.
acceleration
acceleration
time
time
acceleration
acceleration
acceleration
time
time
time

23. 13) α vs t
Given the angular acceleration versus time graph to the right, what is the change in angular velocity of the object?
A) 2/5 rad/s
B) 20 rad/s
C) 40 rad/s
D) None of these
E) Cannot Determine 8 6 4
Acceleration (m/s2) 2
t (s) 2 4 6 8 10 12 14 16 -2 -4 -6 -8