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Rotational MotionPhysics 1 w/ CalculusChapters 9,10, & 11
Rotational Motion
• Think in terms of instead of xθ
≡ddt
≡ddt
Angular Speed
• Think: • Instead of:
ωvx θ
a ≡ ddt
α ≡ddt
Angular Acceleration
• Angular: • linear:
ωv
Shocking SimilarityLinear Motion
Δx
v ≡ΔxΔt
a ≡ΔvΔt
Δx = vot +12at 2
vf2 = vo
2 + 2 ⋅a ⋅d
Angular MotionΔθ
ω ≡ΔθΔt
α ≡ΔωΔt
Δθ =ωot +12αt 2
ω f2 =ωo
2 + 2 ⋅α ⋅ Δθ
Translations = rΔθv = rωa = rα
Example• A car initially traveling at 29 m/s undergoes a
constant negative acceleration of magnitude 1.75 m/s2 after its brakes are applied.
• How many revolutions does each tire make before the car stops? (r = 0.330m)
• What is the angular speed of the wheels when the car has traveled half the total distance?
Example• A grasshopper sits on the edge of a record player
(r = 20 cm). The owner turns the player on to 45 rev/min. It takes 3 seconds to get up to speed.
• What is the angular acceleration of the player?
• What is the tangential acceleration of the grasshopper?
• What is the centripetal and total acceleration?
• If the grasshopper lets go after 5 seconds what will be the launch speed of the grasshopper?
More Similarities
Linear Motion
F∑ = maP = mv
F∑ =dPdt
P = Fiv
KE =12mv2
Angular Motion
τ∑ = IαL = Iω
τ∑ =dLdt
P = τ iω
KEθ =12Iω 2
Torque
τ = r × F
Cross Product
ℓ" × m# =
n"; if cyclic
-n"; if anti-cyclic0; otherwise
⎧
⎨⎪⎪
⎩⎪⎪
Example
• Find the torque on the pipe.
r = 2i! +13 j!( ) inches
F = 5i! − 7 j!( ) pounds
zx
y
r
F
τ = −79k! inch-pounds τ
Notice
• Cross product yields a vector that is equal in magnitude to the area formed by the parallelogram of the other two vectors, and is perpendicular to the plane formed by these two vectors.
r
F
τ
More Similarities
Linear Motion
F∑ = maP = mv
F∑ =dPdt
P = Fiv
KE =12mv2
Angular Motion
τ∑ = IαL = Iω
τ∑ =dLdt
P = τ iω
KEθ =12Iω 2
I = Moment of InertiaAngular equivalent of mass ( )
I ≡ r2 dm = miri2
i∑∫
Moment of Inertia• Like mass, I depends on mass
• Unlike mass, I also heavily depends on position of that mass
I ≡ r2 dm = miri2
i∑∫ Parallel Axis Theorem
I = Icm + mh2
Example
• You wish to rotate a system of particles (all particles have the same mass, m) as shown. What is the moment of inertia of the system?
Example• You wish to rotate a uniform thin rod (of
length L) around it’s center of gravity. Find it’s moment of inertia.
• You now wish to spin a uniform thin rod (m=2Kg, and length=0.5 m) around a point that is 20 cm from the end of the rod. What is it’s new moment of inertia?
Common Objects
Dynamics
A yo-yo has a mass of 0.123 Kg, and the string wraps around the shaft at a radius of 1.1 cm. The yo-yo can be approximated as two solid disks of radius 5 cm. If the yo-yo is allowed to unwind, how fast will it accelerate?
Quick Problem
• Jim’s mass is 60 kg and I is 1.88 kg m2 (let’s neglect his arms). Each weight can be thought of as a point mass, and has a mass of 2 Kg (the arms are 80 cm long). After you spin him at 4 rpm he brings the weights in to his chest (r=5cm). What is his new angular speed? How would this change if his arms were
Example
• The disk has a mass of 8 Kg while the block has a mass of 4 Kg. The pulley at the top is ideal. If the system starts from rest, how many degrees has the disk turned in 15 seconds?
Planets and Gravity• Newton’s law of Gravity:
• Assume circular motion
Fg = −Gm1m2
r2 r!
G = 6.67x10−11 Nm2
Kg2
Example
• The moon’s mass is , the earth’s mass is , and the two planets are approximately apart. What is the approximate orbital period of the moon?
7.36x1022 Kg5.98x1024 Kg
384,000 Km
Momentum and Energy
L = r × p = Iω
KEθ =12Iω 2
Example
• Which rolls faster down an inclined plane, a hoop or a disk or a solid sphere? Assume they all have the same mass and and radius
Nebular Hypothesis
• Sun formed ~5 Billion years ago when gravity pulled large quantities of already rotating gasses from vast distances. The rest of the mass formed into planets.
Test: Does the Nebular Hypothesis Work?
• Find the angular speeds and total angular momentum (orbital + spinning) of Jupiter.
Test: Does the Nebular Hypothesis Work?
ISwirling gassesωo =
Lo = Lf
LplanetsSolar System∑
If...
Lo = Lf
LplanetsSolar System∑
• If we assume the entire mass of the solar system was in a uniform disk slightly larger than the solar system...
⇒ω swirling gas =
Iswirling gas
ω swirling gas
...then
ω swirling gas = 1.33 10−6( ) rad/s
Test: Does the Nebular Hypothesis Work?
MSolar System = 1.99401 1030( ) kg
MSun = 1.99134 1030( ) kg
99.87%
If...• If we assume 99.87% of this initial angular
momentum went into the sun...
Lo = Lf ⇒ωSun should be =0.98 ⋅ LSwirling gas
ISun
ωSun should be
...Then
ωSun should be= 8.40 10−5( ) rad/s
Test: Does the Nebular Hypothesis Work?
ωSun actual = 2.69 10−6( ) rads
31x’s slower than it ‘should’ be
= 8.40 10−5( ) rad/sωSun should be
Test: Does the Nebular Hypothesis Work?
LSolar System = 3.24 1043( ) kg m2
LSun = 0.104 1043( ) kg m2
3.21%
Quotes• Dr. Hurbert Reeves: “The clouds are too hot,
to magnetic, and they rotate too rapidly”
• “The argument (nebular hypothesis) is highly speculative, and some of it borders on science fiction”
• Dr. Stuart Ross: “The ultimate origin of the solar system’s angular momentum remains obscure”