LECTURE 1- ROTATION. Phys 124H- Honors Analytical Physics IB Chapter 10 Professor Noronha-Hostler

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1 LECTURE 1- ROTATION Phys 124H- Honors Analytical Physics IB Chapter 10 Professor Noronha-Hostler

2 CLASS MATERIALS Your Attention (but attendance is OPTIONAL) i-clicker OPTIONAL- EXTRA CREDIT ONLY Homework NOT OPTIONAL- Wiley+ Book (obtained from Wiley+) "Fundamentals of Physics" by Holliday, Resnik and Walker Non-graphing calculator

3 GRADES Done via GRADEBOOK: 224&semester=spring2019&Submit=Login

4 CONFLICTING EXAMS Contact the instructor of the LARGER exam (typically not this class). Make sure to contact instructors well ahead of time. Typically this is only an issue for the final. Watch for snow days!

5 COURSE COMMUNICATION Website: Official Communication: Stuck on a problem? TA first. Wiley+ problem? Contact tech support (always available) to me (patience required). Short, polite s. title: SPRING19 124H ID#

6 OFFICE HOURS Prof. Noronha-Hostler Friday 10am-11am Serin W213 Travis Dore Tuesday 3:30pm-4:30pm ARC 218 Aidan Zabalo Wednesday 9am-10am ARC 238

7 RECITATIONS H1 (Monday 1:40-3:00): 24 H2 (Monday 3:20-4:40): 18 H5 (Monday 8:10-9:30): 5 H8 (Friday 1:40-3:00): 19 Small class size in H5, would receive more individual attention!

8 HOMEWORK DUE DATE? Thursday 11:30pm Sunday 11:30pm

9 SEMESTER OBJECTIVES Describing motion of rotating objects (wheels, tops, spins etc) Describing motion of oscillations/waves (spring, sound waves etc) Describing motion of fluids (water, traffic, galaxies! etc) Understanding temperature/heat Simple machines (fridges/engines)

10 EXPECTATIONS You have read the chapter before class You do your homework You reach out to a TA or myself during office hours, if you need further help If you attend class, that you participate

11 ROTATION

12 TODAY S OBJECTIVES Understand angular motion, angular velocity, and angular acceleration Equations of motion with constant angular acceleration Kinetic Energy and Rotational Inertia Parallel-Axis Theorem Torque Newton s 2nd Law (with Rotation)

13 DEFINITIONS

14 ANGULAR POSITION

15 VELOCITY? ACCELERATION? Angular velocity ω = dθ dt Angular acceleration α = dω dt Are these vector quantities?

16 ANGULAR VELOCITY? ANGULAR ACCELERATION? Angular velocity ω = dθ dt Angular acceleration α = dω dt Are these vector quantities? Yes, they can either be positive or negative to indicate the direction. Note that clocks are negative i.e. clockwise is the negative direction, counterclockwise is positive.

17 RIGHT HAND RULE v = ω r It rotates around an axis, doesn t move in a direction.

18 EXAMPLE FIND THE ANGULAR VELOCITY Given the angular acceleration α(t) = b cos( ft) Find the angular velocity. Note that ω(t 0 = 0) = g

19 FIND THE ANGULAR DISPLACEMENT Given the angular acceleration α(t) = b cos( ft) Find the angular velocity. Note that ω(t 0 = 0) = g α = dω dt ω(t) = dt b cos( ft) dtα(t) = dω(t) = ω(t) ω(t) = b dt cos( ft) = b d sin( ft) + C

20 FIND THE ANGULAR DISPLACEMENT Apply the boundary condition ω(t 0 = 0) = g ω(0) = g = C ω(t) = b d sin( ft) + g

21 WHEN DOES IT REACH ITS MINIMUM ANGULAR VELOCITY? α(t) = b cos( ft) ω(t) = b d sin( ft) + g At the minimum dω(t) dt dω(t) dt = α(t) = 0 = b cos( ft) = 0 Either b = 0 or cos( ft) = 0. When is cos( ft) = 0?

22 WHEN DOES IT REACH ITS MINIMUM ANGULAR VELOCITY? When is cos( ft) = 0? ft = nπ 2 t = nπ 2f Repeats! This means there are multiple minima What sort of movement would have multiple minima in the angular velocity?

23 VELOCITY VS RADIUS Where do you do you sit on a merry-go-round with ω = const, to have the maximal velocity? v = ωr Wheel Demo

24 VELOCITY V. ANGULAR VELOCITY s = θr ω z y v ds dt =v = dθ dt =ω r r θ s x v = ωr

25 DERIVING THE PERIOD Period T Equation of Motion θ θ 0 = ω 0 t αt2 Assuming no angular acceleration α = 0 θ θ 0 = Δθ = ω 0 t t = Δθ ω 0 Period means one rotation i.e. T = 2π ω = 2πr v Δθ = 2π

26 ACCELERATION Tangential component dv dt =a t = dω dt =α r a t ω z y r θ v a r a t s x a t = αr Radial Component a r = v2 r a r v = ωr a r = ω 2 r

27 ANGULAR VS. LINEAR t a r = ω 2 r

28 KINETIC ENERGY+ROTATION

29 KINETIC ENERGY Recall K = 1 2 i m i v 2 i since v = ωr K = 1 2 i (m i r 2 i )ω2 K = 1 2 ω2 i (m i r 2 i )

30 KINETIC ENERGY Recall K = 1 2 i m i v i since v = ωr K = 1 2 i (m i r 2 i )ω2 K = 1 2 ω2 i (m i r 2 i ) What is this? Let s call it I = i m i r 2 i

31 MOMENTUM OF INERTIA Measurement of how difficult it is to rotate an object around a specific axis I = i m i r 2 i Small I Big I K = 1 2 Iω2

32 MOMENT OF INERTIA OF REAL OBJECTS Define axis of rotation I = r 2 dm

33 COMMON OBJECTS (DEMO)

34 PARALLEL-AXIS THEOREM Calculate moment of inertia around an axis shifted by a distance h from the center of mass h com I = I com + Mh 2 Derivation in book

35 FIND MOMENTUM OF INERTIA L com Find the moment of inertia at the center of mass of a barbell of length, L. Given the weights on the edge are mass, M, and the barbell itself has a mass, m.

36 FIND MOMENTUM OF INERTIA L I com = I bar + 2I w com Rod rotating about center I bar = 1 12 ml2 Assume weights are solid disks rotating about its y axis. R I disk = 1 4 MR Ml2 l

37 FIND MOMENTUM OF INERTIA L But the disks are L/2 from the center of mass. Use parallel-axis theorem! com I w = I disk + M ( L 2 ) 2 R I w = 1 4 MR Ml ML2 l

38 FIND MOMENTUM OF INERTIA L I com = I bar + 2I w R com I bar = 1 12 ml2 I w = 1 4 MR Ml ML2 l I com = 1 12 ml2 + 2 [ 1 4 MR Ml ML2 ]

39 TORQUE (DEMO) Torque- turning/twisting action on a body due to a force τ = rf = r F = rf sin ϕ Are you talking about me?

40 NEWTON S 2ND LAW Prove F = ma τ = Iα τ = rf, F = ma t = ma t r, a t = αr = mαr 2, I = mr 2 = Iα Torque needs an acceleration

41 TORQUE OF MERRY-GO-ROUND Sam starts pushing a merrygo-round with a radius R and mass M. Give that the angular velocity is ω(t) = t 1/2 [m/s] What is the torque?

42 TORQUE OF MERRY-GO-ROUND Torque of the merry-go-round τ = Iα Assume it is a round disk rotating around its center. I disk = 1 2 MR2

43 TORQUE OF MERRY-GO-ROUND Find the angular acceleration α = dω dt = d dt ( t 1/2 ) α = t

44 TORQUE OF MERRY-GO-ROUND τ = Iα I disk = 1 2 MR2 α = t τ = 1 2 MR t

45 WORK-KINETIC ENERGY THEOREM W = Fds, s = rθ W = Fd(rθ) here r=const W = Frd(θ) If F=const τ = Fr Recall Work-Kinetic Energy theorem W = τd(θ) = τ ( θ f θ i ) ) W = K f K i = τδθ

46 POWER Using the previous slide P = dw dt = τω Power is a scalar These are both vector quantities P = τ ω

47 PULLEY (DEMO) M 1 Revising Newton s 2nd Law F net = m a M 2 τ = I α Start with drawing free-body diagrams.

48 PULLEY (FREE BODY DIAGRAM) M 2 T M 2 ( a) = T M 2 g F g = mg Still need the tension.

49 PULLEY (FREE BODY DIAGRAM) M 1 R T Torque from forces τ = RF = RT Torque from moment of inertia τ = Iα I disk = 1 2 M 1 R2 Recall clockwise is negative! τ = 1 2 M 1 R2 ( α) RT = 1 2 M 1 R2 α a t = αr T = 1 2 M 1 a

50 COLLECTING THE EQUATIONS M 2 ( a) = T M 2 g T = 1 2 M 1 a a = 2M 2 g M 1 + 2M 2 M 2 ( a) = 1 2 M 1 a M 2 g 1 2 M 1 a + M 2 a = M 2 g If M 1 0 (what you learned last semester) a = g

51 NEXT WEEK Torque as a vector quantity Angular momentum Rolling motion Make sure to get signed up for Wiley+!

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