Conservation of Angular Momentum
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1 Physics 101 Section 3 March 3 rd : Ch. 10 Announcements: Monday s Review Posted (in Plummer s section (4) Today start Ch. 10. Next Quiz will be next week Test# (Ch. 7-9) will be at 6 PM, March 3, Lockett-6
2 Chap. 10: Rotation
3 Conservation of Angular Momentum Describe their motion: θ(t) ω(t) α(t) I Iω Torque
4 Rotation In this chapter we will study the rotational motion of rigid bodies about a fixed axis. To describe this type of motion we will introduce the following new concepts: -Angular displacement θ=s/r - Average and instantaneous angular velocity (symbol: ω ) - Average and instantaneous angular acceleration (symbol: α ) - Rotational ti inertia, also known as moment of inertia (symbol I ) - Torque (symbol τ ) We will also calculate the kinetic energy associated with rotation, write Newton s second law for rotational motion, and introduce the workkinetic energy theorem for rotational motion.
5 Rotation Variables Here we assume (for now): Rigid body - all parts are locked together ( e.g. not a cloud) Axis of rotation - a line about which the body rotates (fixed axis) Reference line - rotates perpendicular to rotation axis θ(t) ( ) = s arc length = r radius looking down Rotation axis Angle is in units of radians (rad) : θ(t) Δθ(t) Δθ = θ θ 1 ω(t) = dθ(t) dt α(t) = dω(t) dt 1 rev = 360 = πr = π rad r = d θ(t) dt ( ) + x ˆ Angular position (displacement) of line at time t: (measured from : ccw) ω(t) Angular velocity of line at time t: (positive if ccw) All points move with same angular velocity Units of rad/sec α(t) Angular acceleration of line at time t: (positive if ccw) All points move with same angular acceleration Units of rad/sec
6 Analogies Between Translational and Rotational Motion Translational Motion Rotational Motion x θ v a v= v + at ω α ω = ω + αt 0 0 at αt x= x o + v 0t+ θ = θ 0 + ω 0t + v v = a x x ω ω = α θ θ 0 ( ) ( ) o mv Iω K = K = m I F = ma τ = Iα F τ P= Fv P = τω 0 0
7 Problem 10-8: The wheel in the picture has a radius of 30cm and is rotating at.5rev/sec. I want to shoot a 0 cm long arrow parallel to the axle without t hitting an spokes. (a) What is the minimum speed? (b) Does it matter where between the axle and rim of the wheel you aim? If so what is the best position. The arrow must pass through the wheel in a time less than it takes for the next spoke to rotate 1 rev Δ t= s.5 rev / s = (a) The minimum speed is v mn = 0cm 0.05s = 400cm / s = 4m / s (b) No there is no dependence upon the radial position.
8 Are angular quantities Vectors? Some! Once coordinate system is established for rotational motion, rotational quantities of velocity and acceleration are given by right-hand hand rule (angular displacement is NOT a vector) Note: the movement is NOT along the direction of the vector. Instead the body is moving around the direction of the vector
9 Sample Problem 10-1: The disc in the figure below is rotating about an axis described by: θ(t) = t + 0 5t bou s desc bed by: θ(t) = t + 0.5t Rewrite the equation θ(t) = ( t 1.) t 0 = 1.s ω = 0rad / s θ( ) = 1.rad θ(0) = 1.0rad θ(1.) = 1.36rad ω(t) = dθ(t) dt =+0.50(t 1.)rad /s α(t) = dω(t) dt =+0.50rad /s
10 Rotation with Constant Rotational Acceleration 1-D translational motion v = v 0 + at x = x 0 + v 0 t + 1 at Rotational motion ω = ω 0 + αt θ = θ 0 + ω 0 t + 1 αt Problem: A disk, initially rotating at 10 rad/s, is slowed down with a constant acceleration of magnitude 4 rad/s. a) How much time does the disk take to stop? ( ω = ω 0 + αt t = 0 10rad/s ) 4rad /s = 30s b) Through what angle does the disk rotate during that time? θ = θ 0 + ω 0 t + 1 αt Δθ = (10rad /s)(30s) + 1 ( 4rad/s )(30s) =1800rad = 86.5rev
11 Equations for constant Angular Acceleration Remember these? Solve Remember these? Solve angular problems the same way.
12 Checkpoint : In four situations, a rotating body has angular position give by the following. To which situations do the angular equations for constant acceleration apply? ( a): θ( t) = 3t-4 θ = (): b () t -5t 4t 6 4 (): c θ() t = t t ( d ): θ () t = 5 t 3
13 Relating Translational and Rotational Variables Rotational position and distance moved s = θ r (only radian units) Rotational and translational speed dr ds d θ v = = = r dt dt dt v = ω r Relating period and rotational speed [ distance= rate time] T = πr v = π units of time (s) ω ωt T = π
14 Relating Translational and Rotational Variables acceleration is a little tricky Rotational and translational lacceleration a) from v = ω r dv dt = d ω dt r a t = d ω dt r = α r from change in angular speed tangential acceleration b) from before we know there salsoa radial a radial component a r = v = ω r r radial acceleration c) must combine two distinct rotational accelerations a tot = a r + a t = ω r + αr
15 Problem 10-30: Wheel A of radius r A =10 cm is coupled by belt B to wheel C of radius r C =5 cm. The angular speed of wheel A is increased from rest at a constant rate of 1.6 rad/s. Find the time needed for wheel C to reach an angular speed of 100 rev/min. If the belt does not slip the tangential acceleration of each wheel is the same. rα = r α α A A C C r A C α A rc = = rad / s The angular velocity of Cis is. t o t ω C C C ω C = α C s = = αc raα A ω r with ω = 100 rev/sec t = 16 s
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