1. Rotational Motion Angles, Angular Velocity and Angular Acceleration

2. Angular Displacement Three different measures of angles: Degrees
Revolutions (1 rev. = 360 deg.)
Radians (2p rads = 360 deg.)

3. Angular Displacement, cont.
Change in distance of a point:

4. Example
An automobile wheel has a radius of 42 cm. If a car drives 10 km, through what angle has the wheel rotated?
a) In revolutions
b) In radians
c) In degrees

5. Solution = 3789 revolutions 2π radians x = 3789 revolutions
Note distance car moves = distance outside of wheel moves
a) Find N: Known: ∆d = 10,000 m; r = 0.42 m
b) Find ∆q in radians
c) Find q in degrees
= 3789 revolutions
2π radians x =
3789 revolutions
1 revolution
∆q = 2.38 x 104 rad.
360 degrees
2.38 x 104 radians x =
2π radians
∆q = 1.36 x 106 deg

6. Angular Speed Can be given in Linear Speed at r Revolutions/s
Radians/s --> Called w
Degrees/s
Linear Speed at r
Angular Speed

7. Example
A race car engine can turn at a maximum rate of rpm. (revolutions per minute).
a) What is the angular velocity in radians per second.
b) If helipcopter blades were attached to the crankshaft while it turns with this angular velocity, what is the maximum radius of a blade such that the speed of the blade tips stays below the speed of sound. (343 m/s)

8. Solution = 1256 radians/s = 0.27 m
a) Convert rpm to radians per second
= 1256 radians/s
b) Known: v = 343 m/s, w = 1256 rad/s
Find r
= 0.27 m

9. Angular Acceleration Denoted by a w must be in radians per sec.
Units of angular acceleration are rad/s²
Every portion of the object has same angular speed and same angular acceleration

10. Analogies Between Linear and Rotational Motion
Linear Motion

11. Linear movement of a rotating point
Distance
Speed
Acceleration
Different points on the same object have different linear motions!
Only works when q, w and a are in radians!

12. Example
A pottery wheel is accelerated uniformly from rest to a rate of 10 rpm in 30 seconds.
a.) What was the angular acceleration? (in rad/s2)
b.)How many revolutions did the wheel undergo during that time?

13. Solution α = 0.0349 rad/s2 1 revolution 15.7 radians x = 2.5 rev
First, find the final angular velocity in radians/s.
a) Find angular acceleration
α = rad/s2
b) Find number of revolutions:
Known wi=0, wf =1.047 rad/s, and t = 30 s
1 revolution
15.7 radians x =
2.5 rev
2π radians

14. Example
A coin of radius 1.5 cm is initially rolling with a rotational speed of 3.0 radians per second, and comes to a rest after experiencing a deceleration of a = rad/s2.
a.) Over what angle (in radians) did the coin rotate?
b.) What linear distance did the coin move?

15. a) Find Dq, Given wi= 3.0 rad/s, wf = 0, a = -0.05 rad/s2
= 90 radians
b) Find the distance the coin rolled Given: r = 1.5 cm and Dq = 90 rad
= 135 cm

16. 1) 2) ωf2 = 4.22 + 2(22.4)(18.8) Given: ωi = 0.0 rad/s ωf = 4π rad/s
4π = 0.0+ (α)(0.5)
t = 0.5 sec
α = ? rad/s2
α = 25.1 rad/s2
Given: 2)
ωi = 40 rpm
α = 22.4 rad/s2
∆θ = 3 rev
ωf = ? rad/s
ωf2 = (22.4)(18.8)
4.2 rad/s
18.8 rad
ωf = 29.3 rad/s

17. 3) ∆θ = (0.0)(2.5) + ½(0.35)(2.5)2 Given: α = 0.35 rad/s2
a = 0.18 m/s2
0.18 = (0.35)(r)
ωi = 0.0 rad/s
r = 0.51 m
t = 2.5 sec
∆θ = (0.0)(2.5) + ½(0.35)(2.5)2
∆θ = 1.09 rad
0.17 revolution

18. 4) 2∆θ = (0 + 5.2)(6.0) Given: r = 0.5 m m = 100.0 kg
0 = (α)(6.0)
ωi = 50 rpm
t = 6.0 sec
α = rad/s2
ωf = 0 rpm
α = ? rad/s2
∆θ = ? rotations
2∆θ = ( )(6.0)
∆θ = 15.6 radians
= 5.2 rad/s
2.5 rotations

19. 5) convert to rad/s Given: ωi = 10 rpm t = 4.5 sec
15.2 = (α)(4.5)
ωf = 145 rpm
α = ? rad/s2
α = 3.14 rad/s2
convert to rad/s
ωi = 1.05 rad/s
ωf = 15.2 rad/s