PHYSICS 231
Lecture 15: Rotations
Remco Zegers Walk-in hours Thursday 11:30-13:30
Helproom
PHY 231
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Radians & Radius
Circumference=2πr
Part s=θr
r
s
θ
θ=s/r
θ in radians!
360o=2π rad = 6.28… rad
θ (rad) = π/180o θ (deg)
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Angular speed and acceleration
θ f −θi
∆θ Average angular
ϖ=
=
∆t velocity
t f − ti
∆θ
ω = lim
∆t → 0 ∆t
Instantaneous
Angular velocity
Angular velocity : rad/s
rev/s
rpm (revolutions per
minute)
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example
What is the angular velocity of earth around the sun?
Give in rad/s, rev/s and rpm
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Angular acceleration
Definition: The change in angular velocity per time unit
ω f − ωi
∆ω Average angular
α=
=
t f − ti
∆t acceleration
∆ω
α = lim
∆t →0 ∆t
Instantaneous angular
acceleration
Unit: rad/s2
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Equations of motion
Linear motion
Angular motion
X(t)=x(0)+v(0)t+½at2
θ(t)= θ(0)+ω(0)+t½αt2
V(t)=V(0)+at
ω(t)= ω(0)+αt
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example
θ=0
A person is rotating a wheel. The
handle is initially at θ=90o. For 5s
the wheel gets an constant angular
acceleration of 1 rad/s2. After that
The angular velocity is constant.
Through what angle will the wheel
have rotated after 10s.
ω
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Angular
∆s
∆θ =
r
∆θ 1 ∆s
=
∆t r ∆t
v
ω=
r
v = ωr
Linear velocities
V
V is called the tangential velocity
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Angular
v = ωr
∆v = ∆ω ⋅ r
∆v ∆ω
r
=
∆t
∆t
a = αr
linear acceleration
velocity
Change in velocity
Change in velocity per time unit
acceleration
The linear acceleration equals the angular acceleration
times the radius of the orbit
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a
example
b
5m
The angular velocity of a is 2 rad/s.
a) What is its tangential velocity?
If b is keeping pace with a,
b) What is its angular velocity?
c) What is its linear velocity?
6m
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A rolling coin
d
d
ω0=18 rad/s
α=-1.80 rad/s
d=0.02 m
a) For how long does the coin roll?
b) What is the average angular velocity?
c) How far does the coin roll before coming to rest?
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gears
If ω1=3 rad/s, how fast
Is the bike going?
b) What if r1=0.1 m ?
r3=0.7 m
r1=0.3
r2=0.15
b)
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