Chapt. 10 Rotation of a Rigid Object About a Fixed Axis

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1 Chapt. otation of a igid Object About a Fixed Axis. Angular Position, Velocity, and Acceleration Translation: motion along a straight line (circular motion?) otation: surround itself, spins rigid body: no elastic, no relative motion rotation: moving surrounding the fixed axis, rotation axis, axis of rotation Angular position: s o θ, rev 6 π ( rad), r 6 rad 57. π Angular displacement: θ θ θ An angular displacement in the counterclockwise direction is positive Angular velocity: averaged instantaneous: θ θ θ ω t t t dθ ω, rpm rev per minute dt Angular acceleration: averaged instantaneous: Sample Problem: dω α dt α ω t The disk in Fig. -5a is rotating about its central axis like a merry-go-round. The angular position θ(t) of a reference line on the disk is given by θ..6t.5t with t in seconds, q in radians, and the ero angular position as indicated in the figure. a) Graph the angular position of the disk versus time from t -. s to t 6. s. Sketch the disk and its angular position reference line at t -. s, s, and 4. s, and when the curve crosses the t axis. b) At what time t min does θ(t) reach the minimum value? What is that minimum value? c) Graph the angular velocity ω of the disk versus time from t -. s to t 6. s. Sketch the disk and indicate the direction of turning and the sign of ω at t -. s and 4. s, and also at t min. o

2 ight-hand rule Specify Your Axis. otational Kinematics: The igid Object Under Constant Angular Acceleration Sample Example: A grindstone rotates at constant angular acceleration α.5 rad/s. At time t, it has an angular velocity of ω -4.6 rad/s and a reference line on it is horiontal, at the angular position θ. d ω dt α.5rad / s, d. 5dt dθ ω 4.6.5t, dt ω, ω ω.5 t, ω 4.6.5t θ θ 4.6t. t, 75 θ 4.6t.75t a) At what time after t is the reference line at the angular position θ 5. rev? c) At what time t does the grindstone momentarily stop?

3 . Angular and Translational Quantities The position: s rθ The speed (velocity?): The acceleration: Sample Example: v rω, a t rα, πr T v v a r rω r Figure shows a centrifuge used to accustom astronaut trainees to high accelerations. The radius r of the circle traveled by an astronaut is 5 m. (a) At what constant angular speed must the centrifuge rotate if the astronaut is to have a linear acceleration of magnitude g? 9.8 r ω 5 ω 9.8, ω.68rad / s 5 (b) What is the tangential acceleration of the astronaut if the centrifuge accelerates at a constant rate from rest to the angular speed of (a) in s? ω.68 at rα rα avg 5 5.4m / s.4 otational Kinetic Energy K mivi miω ri m iri m r (rotational inertia), i i K ω Example: The Oxygen olecule Consider an oxygen molecule (O ) rotating in the xy plane about the axis. ω rotation axis passes through the center of the molecule, perpendicular to its length. The mass of each oxygen atom is.66 x -6 kg, and at room temperature the average separation between the two atoms is d. x - m. particles.) The (The atoms are modeled as (a) Calculate the moment of inertia of the molecule about the axis. m i r i what is the radius i r? d m d m (b) f the angular speed of the molecule about the axis is 4.6 X rad/s, what is its rotational kinetic energy? mv Ek -> E ω Example: Four otating Object m a b m y x

4 Calculate the moment of inertia if the rotation is about the axis and about the y axis. what is the radius i m i r i r? ma ma b b y b b.5 Calculation of oments of nertia Cartesian Coordinate: Spherical Coordinate: x, y, r,θ, φ r cosθ x r sinθ cosφ, y r sinθ sinφ x y Cylindrical Coordinate: x r cosθ, y r sinθ r,θ, x y (a) πρ ρ θ d (b) π ( )ρ π r rρdθdr πρr dr ( ) (c) π ρ r π rρ dθdrd π r dr ρ (d) 4

5 π ρ π ( r sin θ ) ρrdθdrd 4 (e) from (d), (f) <<, π 4 ρ π π r sin θρ r sinθdφrdθdr 5 (g) π σ 4π ( sinθ ) π sinθdθ ( sin ) sin d ( cos ) d( cos ) θ π θ θ θ θ π (h) πρ θρ θ sin d (i) b / a / ( ) x y dxdy ( a b ) ab b / a / Parallel axis theorem: h r dm [( x a) ( y b) ]dm ( x y ) dm a xdm b ydm ( a b ) dm cm h 5

6 Sample Example: (a) What is the rotational inertia com of this body about an axis through its center of mass, perpendicular to the rod as shown? i i m m m r m (b) What is the rotational inertia of the body about an axis through the left end of the rod and parallel to the first axis? m r i i m m m use parallel axis theorem: m m m Example: otating od A uniform rod of length and mass is free to rotate on a frictionless pin through one end. The rod is released from rest in the horiontal position. (a) What is the angular speed of the rod at its lowest position?.. cm, ω g, gh ω cm ω, ( ), Energy conservation: gh cm v cm.6 Torque g ω, v cm ω r r Torque is a vector: N τ F rf sinφ, τ τ τ τ... Torque <-> Force, What is the difference? Torque can increase or decrease the angular velocity. What about force? Example: The net torque on a cylinder r τ ( T T )kˆ 6

7 .7 The igid Body Under a Net Torque F t Ftan gent, F r F along _ the _ radius F t F is used for angular acceleration tan gent F ma mrα, τ rf t mr α α t t The torque acting on the particle is proportional to its angular acceleration. Example: otating od A uniform rod of length and mass is attached to a frictionless pivot and is free to rotate about the pivot in the vertical plane. The rod is released from rest in the horiontal position. What is the initial angular acceleration?, τ g α, pivot g α, a t α g at right end Example: Angular Acceleration of a Wheel A wheel of radius, mass, and moment of inertia is mounted on a frictionless horiontal axle. A light cord wrapped around the wheel supports an object of mass m. Calculate the angular acceleration of the wheel, the linear acceleration of the object, and the tension in the cord., mg T ma a mg ma,, τ T α, a α mg a, m T α a mg m, α 7

8 Example: Atwood achine evisited Two blocks having masses m and m are connected to each other by a light cord that passes over two identical frictionless pulleys, each having a moment of inertia and radius. Find the acceleration of each block and the tension T, T, and T in the cord. (No slipping) g T m a T T m T mg m, ( ) α, ( T T ) α a, α a ( m m ) a m m g.8 Energy Considerations in otational otion r r dw F ds dw τ dθ ( F sinφ) rdθ dw dθ P τ τω dt dt dω dω dθ dω τ α ω dt dθ dt dθ work-kinetic energy theorem: τ d θ ωdω, ωf W ωi ωdω ω f ω i otational motion inear motion W θf τ dθ θi W xf xi F x dx K ω P τω p ω d τ dt K mv P Fv p mv dp F dt 8

9 Example: otating od evisited A uniform rod of length and mass is free to rotate on a frictionless pin passing through one end. The rod is released from rest in the horiontal position. Determine the tangential speed of the and the tangential speed of the lowest point on the rod when it is in the vertical position. gh ω g, g ω, v ω g v end ω g ω, g pivot 4.9 olling otion of a igid Object olling as rotation and translation combined θ v cm ω, ω sinθ iˆ ω cosθ ˆj v edge v edge ω ( sinθ )ˆ i ω cosθ ˆj olling as pure rotation 9

10 .. K v K, pω ω P, ω K ω Sample Problem: A uniform solid cylindrical disk, of mass.4 kg and radius 8.5 cm, rolls smoothly across a horiontal table at a speed of 5 cm/s. What is its kinetic energy K? K ω v v v v 4 4 mg sin θ f s ma, a α α f s, f s mg sinθ a m a Sample Example: A uniform ball, of mass 6. kg and radius, rolls smoothly from rest down a ramp at angle θ.. (a) The ball descends a vertical height h. m to reach the bottom of the ramp. What is its speed at the bottom? v ω gh, 5 ω, v v gh v gh, v 4.m / s 5 7 (b) What are the magnitude and direction of the friction force on the ball as it rolls down the ramp?

11 g sinθ 4.9 a.5m / s 5, f s a a 8. 4N 5

PHYSICS 111 HOMEWORK SOLUTION #10. April 8, 2013

PHYSICS 111 HOMEWORK SOLUTION #10. April 8, 2013 PHYSICS HOMEWORK SOLUTION #0 April 8, 203 0. Find the net torque on the wheel in the figure below about the axle through O, taking a = 6.0 cm and b = 30.0 cm. A torque that s produced by a force can be