1. Angular Acceleration
Objectives:
• Introduce the concept of angular acceleration
• Learn how to compute and estimate angular
acceleration
• Understand the difference between average
and instantaneous angular acceleration
• Learn to use the laws of constant acceleration
Angular Acceleration
• The rate of change of angular velocity
change in angular
angular velocity
acceleration =
change in time
• Shorthand notation:
ωfinal – ωinitial ∆ω
α = =
tfinal – tinitial ∆t
• Has units of (angular units)/time2
(e.g. radians/s2, °/s2)
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2. Computing Angular Acceleration
• Angular acceleration reflects a change in velocity of
rotation
ω2 – ω1 30°/s
α = = = 20°/s2
t2 – t1 1.5 s
ω2 = 40°/s
ω1 = 10°/s
∆θ
orientation at
t2 = t1+1.5 s
orientation
at t1
axis of rotation
Effects of Angular Acceleration
• Velocity ω and acceleration α in same direction:
magnitude of angular velocity increases
• Velocity ω and acceleration α in opposite direction:
magnitude of angular velocity decreases (deceleration)
Velocity Acceleration Change in Velocity
(+) (+) Increase in + dir.
(+) (–) Decrease in + dir.
(–) (–) Increase in – dir.
(–) (+) Decrease in – dir.
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3. Example Problem #1
A volleyball player spikes the ball.
To bring her arm forward, she begins extending her
shoulder from a flexion angle of 225°.
She contacts the ball 120 ms later with her
shoulder flexed to 160° and extending at 700°/s
100 ms later, at the end of follow-through, her
shoulder stops extending at a flexion angle of
135°
What was the average acceleration at the shoulder
before and after ball impact?
Instantaneous Angular Accel.
• Previous formulas give the average angular
acceleration between initial time (t1) and final time (t2)
• Instantaneous angular acceleration is the angular
acceleration at a single instant in time
• Estimate instantaneous angular acceleration using
the central difference method:
ω (at t1 + ∆t) – ω (at t1 – ∆t)
α (at t1) =
2 ∆t
where ∆t is a very small change in time
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4. Angular Acceleration as a Slope
• Graph of angular velocity vs. time
slope = instantaneous
α at t1
ω (deg/s)
slope = average
α from t1 to t2
∆ω(1→2)
∆t(1→2)
∆t
t1 t2 time (s)
Estimating Angular Acceleration
ω (deg/s)
Identify points with
zero slope = points
with zero acceleration
0 Portions of the curve
time (s) with positive slope
have positive accel.
(i.e. acceleration in
α (deg/s2)
the + direction)
Portions of the curve
with negative slope
0 have negative accel.
time (s)
(i.e. acceleration in
the – direction)
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5. Example Problem #2
Pictured is the absolute angle of a hockey stick during
a slap shot. Sketch the angular velocity and
angular acceleration during the shot.
80
60
40
Stick angle (deg)
20
0
-20
0 0.2 0.4 0.6 0.8 1
θ
-40
-60
Time (s)
Laws of Constant Angular Accel.
• When angular acceleration is constant:
ω2 = ω1 + α * ∆t
∆θ = ω1 * ∆t + (½) α * ( ∆ t)2
ω22 = ω 12 +2 α * (∆ θ )
where:
α = angular acceleration
ω 1 = angular velocity at initial (or first) time t1
ω 2 = angular velocity at final (or second) time t2
∆ θ = angular displacement between t1 and t2
∆ t = change in time (= t2 – t1 )
Use + values for + direction, – values for – direction
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6. Example Problem #3
A discus thrower starts his spin while standing
facing the back of the circle.
He releases the discus 2 seconds later after
completing 1.5 revolutions to his left.
What was his angular acceleration during the throw
(assuming a constant rate of acceleration)?
How fast was he spinning after the first
half-revolution?
How fast was he spinning at the time of release?
Rotation as a Vector
• Rotational quantities ( θ, ω, α ) can be expressed by
vectors directed along the axis of rotation.
• Length of vector indicates magnitude
• Direction of vector determined by “right hand rule”
– Curl fingers of right hand in direction of rotational quantity
– Thumb points in direction of vector
z z
axis of rotation
ω
y y
x x ω
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7. Linear & Angular Motion
(part 1)
Objectives:
• Learn the relationship between linear and
angular distance for a body in rotation
• Learn the relationship between linear and
angular speed for a body in rotation
Linear & Angular Distance
• The curvilinear distance (d) traveled by a point on
a rotating body is:
d
d=rφ
where: φ
• r = radius of rotation
(distance of the point r
from the axis of rotation)
• φ = angular distance
traveled in radians! axis of rotation
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8. Linear & Angular Speed
• The linear speed (s) of a point on a rotating body
is:
d rφ φ
s=rσ s= = =r =rσ
∆t ∆t ∆t
where: d
• r = radius of rotation
• σ = angular speed φ
in radian/s !
r
axis of rotation
Radius of Rotation, Distance & Speed
The greater the radius of rotation (r):
• the greater the curvilinear distance (d) traveled for a
given angular distance ( φ)
• the greater the linear speed (s) for a given angular
speed ( σ )
d1
d=rφ
d2
s=rσ
if r1 > r 2 φ
then d1 > d2 r2 r1
axis of rotation
s1 > s2
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9. Example Problem #1
Two distance runners are racing. Runner #1 is in
the inside lane. Runner #2 is on his shoulder, in
the second lane.
The inside radius of the track is 36.8 m. Each lane
is 1.1 m wide.
How much farther must Runner #2 run on each
turn?
In a 10,000 m race (25 laps), how much farther
would Runner #2 have to run?
Example Problem #2
A baseball player swings a bat at a speed of
250°/s.
If he makes contact with the ball 40 cm from the
axis of rotation of his body, what is the speed of
the bat at the point of impact?
What is the speed at the point of impact if he
makes contact at a distance of 80 cm instead?
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