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”