PEP 294: Lecture Notes
XI. Angular Kinematics
Units: Degree vs. Radian (pp. 333-336, 341-343)
- degree (deg): 1 rev = 360°
- radian (rad): the angle with which the length of the arc becomes the same to the radius
1 rev = 2p rad = 360°
p rad = 180°
p/2 rad = 90°
p/3 rad = 60°
p/4 rad = 45°
1 rad = 57.3 deg
Quantities (pp. 339-345)
- Related Problems: IP1, 2, 3, 4, 5, 7, AP7
1. Angular Position
 | Angular position: |
- angular location at given instance
- direction:
(a) counterclockwise: +
(b) clockwise: -
| Standard unit: rad |
| Example: |
2. Angular Displacement
| Angular displacement: |
- change in angular position, net effect of angular motion (vector quantity)
- direction
(a) counterclockwise: +
(b) clockwise: -
| Standard unit: rad |
| Examples: |
1. Initial angular position = 0.1 rad, final angular position = 0.9 rad => angular displacement?
= 0.9 - 0.1 = 0.8 rad
2. Golf swing: angular displacements of the phases and overall displacement?
Angular positions:
Position 1: q1 = 3p/2
Position 2: q2 = p/4
Position 3: q3 = 3p
Phase 1: backswing
Dq = p/4 - 3p/2 = -5p/4 rad
Phase 2: forward swing
Dq = 3p/2 - p/4 = 5p/4 rad
Phase 3: follow-through
Dq = 3p - 3p/2 = 3p/2 rad
Total: Dq = 3p - 3p/2 = 3p/2 rad
3. Angular Velocity
| Angular velocity: |
- rate of change in angular position
- angular velocity = angular displacement / time
w =
=
| Standard unit: rad/s |
| Examples: |
1. Angular displacement = 0.8 rad, elapsed time = 0.4 s => angular velocity?
Dq = 0.8 rad, Dt = 0.4 s
w =
= 2 rad/s (2 rad/s counterclockwise)
2. Initial angular position = 0.9 rad, final angular position = 0.1 rad, elapsed time = 0.4 s => angular velocity?
q1 = 0.9 rad, q2 = 0.1 rad, Dt = 0.4 s
w =
=
= -2 rad/s (2 rad/s clockwise)
4. Angular Acceleration
| Angular acceleration: |
- rate of change in angular velocity
- angular acceleration = (change in angular velocity) / (elapsed time)
a =
=
| Standard unit: rad/s2 (rad/s/s) |
| Examples: |
1. Golf swing: time from end of back swing to impact = 0.8 s, angular acceleration = 1.5 rad/s2 => angular velocity at impact?
w1 = 0.0 rad/s, a = 1.5 rad/s2, Dt = 0.8 s
1.5 =
=> w2 = 1.2 rad/s
2. If angular acceleration = 0, what happens to angular velocity?
a =
Linear Motion vs. Angular Motion (pp. 347-354)
- Related problems: IP10, AP1, 2, 3, 4, 8, 9, 10
1. Circular Motion
| Velocity relationship: |
- magnitude: linear velocity = (radius) x (angular velocity)
v = rw
(a) for the same w , v is proportional to r.
(b) for the same r, v is proportional to w.
- direction of the linear velocity: tangential
(a) hammer throwing
(b) David vs. Goliath (biomechanics of slinging)
| Example -- Baseball batting: angular velocity = 30 rad/s => velocity of the bat at 20 cm & 40 cm from the handle? |
r = 20 cm or 40 cm
At 20 cm:
v = rw
= (20 cm)(30 rad/s) = (0.2 m)(30 rad/s) = 6 m/s
At 40 cm:
v = rw
= (0.4)(30) = 12 m/s
Therefore, v40 = 2·v20 (v40 : v20 = 40 : 20)
