1. PHSX 114, Monday, October 13, 2003 Reading for today: Chapter 8 (8-1 -- 8-3)Reading for today: Chapter 8 (8-1 -- 8-3) Reading for next lecture (Wed.): Chapter 8 (8-4 -- 8-6)Reading for next lecture (Wed.): Chapter 8 (8-4 -- 8-6) Homework for today's lecture: Chapter 8, question 3; problems 5, 8, 17, 22Homework for today's lecture: Chapter 8, question 3; problems 5, 8, 17, 22

2. Rotation and translation Motion of the center of the object is translationalMotion of the center of the object is translational Motion about the center of the object is rotationalMotion about the center of the object is rotational

3. Rigid body rotating about a fixed axis A "rigid body" doesn't flex or distort as it rotatesA "rigid body" doesn't flex or distort as it rotates "Axis" is the line about which the object rotates"Axis" is the line about which the object rotates If the axis is not moving, can use a single angular variable, θ, to describe the angular positionIf the axis is not moving, can use a single angular variable, θ, to describe the angular position This the rotational equivalent to one-dimensional translational motion (single variable x)This the rotational equivalent to one-dimensional translational motion (single variable x)

4. Angular position Angle θ measured from x-axisAngle θ measured from x-axis Counterclockwise rotation is a positive angular displacementCounterclockwise rotation is a positive angular displacement Clockwise rotation is a negative angular displacementClockwise rotation is a negative angular displacement Most common units: revolutions, degrees, radiansMost common units: revolutions, degrees, radians 1 revolution = 2π radians = 360 degrees1 revolution = 2π radians = 360 degrees angle in radians = arc length subtended/radius; θ = l/rangle in radians = arc length subtended/radius; θ = l/r

5. Angular velocity Angular velocity = angular displacement/ time interval; ω=Δθ/ΔtAngular velocity = angular displacement/ time interval; ω=Δθ/Δt ExampleExample

6. Angular acceleration Angular acceleration = change in angular velocity/ time interval; α=Δω/ΔtAngular acceleration = change in angular velocity/ time interval; α=Δω/Δt If α is constant, α = (ω-ω 0 )/t => (ω-ω 0 ) = αt => ω = ω 0 + αtIf α is constant, α = (ω-ω 0 )/t => (ω-ω 0 ) = αt => ω = ω 0 + αt Other constant α equations: θ = θ 0 + ω 0 t + ½αt 2 ; ω 2 = ω 0 2 + 2α ΔθOther constant α equations: θ = θ 0 + ω 0 t + ½αt 2 ; ω 2 = ω 0 2 + 2α Δθ Compare these to constant a equations: x -> θ; v->ω; a -> αCompare these to constant a equations: x -> θ; v->ω; a -> α Compare these two examplesCompare these two examples

7. Your turn: A merry-go-round is initially at rest and is given a constant angular acceleration of 0.2 rad/s 2. a) What is the merry-go-round's angular velocity after 10 s? b) What angle has it turned through? c) How many revolutions has it made? Answer: a) ω = ω 0 + αt = 0 + (0.2 rad/s 2 )(10 s) = 2 rad/s b) θ = θ 0 + ω 0 t + ½αt 2 = 0 + 0 + ½(0.2 rad/s 2 )(10 s) 2 = 10 rad c) (10 rad)(1 rev/2π rad) = 1.6 rev

8. Relation between angular speed and linear speed For a rotating rigid body, ω is the same at all points on the bodyFor a rotating rigid body, ω is the same at all points on the body Linear speed increases as the radial distance r from the rotation axis increasesLinear speed increases as the radial distance r from the rotation axis increases θ = l/r => Δθ/Δt = Δl/rΔt => ω=v/rθ = l/r => Δθ/Δt = Δl/rΔt => ω=v/r v=rω, linear speed = radius times angular speedv=rω, linear speed = radius times angular speed

9. Relation between angular speed, frequency, and period Recall the period T is the time for one complete revolutionRecall the period T is the time for one complete revolution θ=ωt; (2π radians) = ωT => T = 2π/ωθ=ωt; (2π radians) = ωT => T = 2π/ω frequency, f, is the number of revolutions per secondfrequency, f, is the number of revolutions per second 1 Hertz (Hz) = 1 rev/s1 Hertz (Hz) = 1 rev/s Since T is seconds per revolution, f = 1/TSince T is seconds per revolution, f = 1/T

10. Example relating angular speed, frequency, and period If ω= 3 rad/s, what are T and f? T = (2π rad/rev)/(3 rad/s) = 2π/3 s/rev = 2.1 s/rev; T = (2π rad/rev)/(3 rad/s) = 2π/3 s/rev = 2.1 s/rev; f = 1/T = 1/ (2.1 s/rev) = 0.48 rev/s (Hz)

11. Relation between angular acceleration and linear acceleration α increases the rate of spin, so increases tangential linear velocityα increases the rate of spin, so increases tangential linear velocity From v=rω, Δv/Δt = rΔω/Δt => a tan = rαFrom v=rω, Δv/Δt = rΔω/Δt => a tan = rα Centripetal acceleration: a R = v 2 /r = (rω) 2 /r = rω 2Centripetal acceleration: a R = v 2 /r = (rω) 2 /r = rω 2

12. Relation between angular speed and linear speed for a smoothly rolling object Example: relate the linear speed of a bicycle to the angular speed of the wheelsExample: relate the linear speed of a bicycle to the angular speed of the wheels Circumference of object maps smoothly onto surface (see paper towel roll)Circumference of object maps smoothly onto surface (see paper towel roll) Distance l traveled by bicycle equals distance l traveled by point on edge of wheelDistance l traveled by bicycle equals distance l traveled by point on edge of wheel l=rθ => v=rωl=rθ => v=rω ExampleExample

13. Your turn In the preceding example, what is the tangential and centripetal acceleration at t= 0 s? Answer: as before, a tan = rα = (0.3 m)(4.2 rad/s 2 ) = 1.25 m/s 2 a R = rω 2 = (0.3 m)(50 rad/s) 2 = 750 m/s 2