Rotational Motion Angular Measure (radian) Angular Speed and velocity Centripetal Acceleration Centripetal Force Angular Acceleration
Mar 26, 2015
Rotational Motion
Angular Measure (radian) Angular Speed and velocity Centripetal Acceleration Centripetal Force Angular Acceleration
Angular measurey
x
ry
x
x=r cos()
y=r sin()
Angular measurey
x
r
x
=
2 radians = 360°
s
s
rradians
1.0 radian = 57.3°
How far to the moon?
Early Greek scientists estimated that the earth’s diameter is 3.5 times larger than the moon’s.
If a penny covers up the moon when held 200 cm from your eye, How far is the moon from Earth? (the diameter of a penny is 1.9 cm)
Small angles
rh s
h≈ s & r ≈ x
Sin ≈ ≈ tan h/r ≈ s/r ≈ h/x
x
Angular speed and velocity
Average angular speed
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ϖ =Δt
θ = ϖt (θ i = 0)
ω =dθdt
Linear speed and angular speed
rh s
x
v =st=
rt
=rϖ
v=rω
Linear speed and angular speed
rh s
Sin ≈ ≈ tan h/r ≈ s/r ≈ h/x
x
v =rω
Angular acceleration
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α =ΔωΔt
ω = ωi +α t
Angular acceleration
α =Δv
rt
=Δvt ⋅r
=a t
ra t = rα
€
α =ΔωΔt
ω = ωi +α t
Circular Motion(with constant α
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ϖ =(ω + ωi )
2
Circular Motion(constant α
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ω =ω i +α t
θ =θ i +ω f +ω i
2t
θ =θ i +ωi t +12
α t2
ω2 = ωi2 + 2α (θ −θ i )
Kinetic Energy
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Ki =12mivi
2 =12miri
2ω2
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K =12miri
2ω2 =∑12
miri2
[ ]ω2 =∑
12Iω2
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K =12Iω2
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I = miri2
[ ] or I = r2dm∫∑
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Iz = Icm +Md2
d
Torque
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τ =rF sinφ = Fd
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τ =τ1 + τ 2 = +F1d1 − F2d2
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τ∑ A= IAα
a=?
T=?
a=?
T=?
F=mamg-T=ma
τ= IαTR = 1/2MR2 (a/R)
T= 1/2M a
mg-T=ma
mg- 1/2M a=ma
mg= 1/2M a+ma
mg= (1/2M +m)a
mg (1/2M +m)
T= 1/2M a
a =
Work and Power
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work =r τ ⋅
r θ
Power =r τ ⋅
r ω