# Chapter 10: Radians Rotational Motion s = rθ ω θ...

### Transcript of Chapter 10: Radians Rotational Motion s = rθ ω θ...

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Chapter 10: Rotational Motion

Brent Royuk Phys-111

Concordia University

Angular Ideas from Chp. 6 • Angular distance: Δθ = θ - θo • Radians

– The radian equation: s = rθ • Angular speed and velocity: • vt = r ω

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ω =Δθ

Δt

Rolling Motion • Rigid Bodies, Translations and Rotations

– A rigid body is a system of particles in which the distances between the particles do not vary.

– Analysis is much simpler than non-rigid body – Forces acting through center of gravity produce

translations. • Tangential speed is equal to translational

speed for a rolling object. • Examples

– Rolling without slipping: If ω = .89 s-1 and r = .085 cm, what is v for the rigid body?

– What about a book on pencils with same ω and r as above?

Rolling Motion • The wheel in the figure is rolling to the right

without slipping. • Rank in order, from fastest to slowest, the

speeds of the points labeled 1 through 5.

Angular Acceleration • Definition:

– Uniform angular acceleration • ω = ωo + αt

– Units – Vector direction

• Tangential Acceleration – at = rα

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α =Δω

Δt

Rotational Kinematics

Translational Rotational s = r x = vt = t vt = r v = vo + at = o + t

at = r

x = vot +12

at 2 θ = ω ot +

12αt 2

v2 = vo

2 +2 ax ω2 = ω o

2 +2αθ

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Rotational Kinematics • Examples

– A potter’s wheel rotates from rest to 210 rpm in a time of 0.75 s. a) What is the angular acceleration of the wheel during this time? b) How many revolutions does the wheel make during this time interval?

– How many revolutions does it take an angularly accelerating object to accelerate up to 2.8 rad/s at a rate of 0.058 rad/s2?

– What is the angular acceleration of a spinning object that accelerates smoothly from “rest” and makes 4.0 complete revolutions while attaining an angular speed of 5.5 rad/s?

Torque • Definition: • Vector notation- a cross product:

– Compare 0o and 90o – Compare pushing door on outside or inside – Direction: right hand rule – Demo gadget

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! τ =! r ×! F τ = rF

⊥= rF sinθ

Torque • Lever Arms

– You can either calculate torque as the product of r and or F and . F⊥ r⊥

Rotational Inertia • What is the rotational equivalent of

mass? – Start with F = ma. – Multiply by r, use a = α r – Result: τ= m r2 α

• I = moment of inertia: – Single particles are easy: add them

up. – Continuous mass distributions are

harder: must use calculus – See Figure 10.12 (next slide)

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I = miri2∑

Moment of Inertia Rotational Dynamics

• Using Newton’s 2nd: • Demo rod • Example: Wrap a string around a 1.5

kg solid sphere with radius 8.0 cm. Pull on the string with a force of 8.0 N. Find the angular acceleration

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! τ NET = I

! α

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Rotational Kinetic Energy • Equation: • Examples

– What is the rotational kinetic energy of a 450-g solid sphere with a diameter of 23 cm rotating at rate of 17 rpm?

– What is the total rotational kinetic energy of a 18-kg child riding on the edge of a merry-go-round of mass 160 kg and r = 2.5 m that is rotating with a period of 2.0 s?

• Rolling Objects – For rolling objects: – Examples

• What percentage of the kinetic energy of a rolling ball is rotational? How about a cylinder?

• Compare the speed of a ball rolling down an inclined plane with its speed if it would slide.

– Demo objects: ring and disk, eyeball, soup cans

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KTOT =12

mv 2 +12

Iω2

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K =12

Iω2 Angular Momentum

– For a single particle, derive L = m v r !L = I !ω

Angular Momentum • Conservation of Angular Momentum

– Lb = La – Ib ωb = Ia ωa – Pulling string while bob rotates – Spinning skaters – Rotating platform

• What if your initial angular speed is 3.74 rad/s, and by pulling your arms in you change your moment of inertia from 5.33 kg m2 to 1.60 kg m2?

– Spacecraft and cats – Jet engines not bolted too firmly in case of “seize-

up.”

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Angular Momentum

Lb = La

Ibωb = Iaωa

Rotational Vectors • Direction of the L-vector • Gyroscopic stability

– Football spirals – Bullets

• Directional L conservation – Bicycle wheel on platform

• Precession – ∆L = τ ∆t – Gyroscope – Diving airplanes – Helicopters – Motorcycle turns

Rotational Vectors • Direction of the L-vector • Gyroscopic stability

– Football spirals – Bullets

• Directional L conservation – Bicycle wheel on platform

• Precession – ∆L = τ ∆t – Gyroscope – Diving airplanes – Helicopters – Motorcycle turns

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The Big Picture Translational Rotational

s = rθ x = vt θ = ωt vt = rω v = vo + at ω = ωo + αt

at = rα

x = vot +12

at 2 θ = ω ot +

12αt2

v2 = vo

2 +2 ax ω2 = ω o

2 +2αθ

! F

! τ =! r ×! F

τ = rF sin θ m

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I = miri2∑

! F NET =

! F ∑ = m ! a

! τ NET = I ! α

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W = Fd cosθ W = τθ

K TOT =

12

mv 2 +12

Iω 2 K =

12 mv 2

K

rot= 1

2Iω 2

! p = m ! v

! L = I ! ω

F Δt = m Δv τ Δt = I Δω

�

Pav = ΔEΔt

P = Fv

P = τω