康涅狄格州大学：《普通物理》（英文版） Lecture 10 displacement

Physics 121, Sections 9, 10, 11, and 12 Lecture 10 Today's Topics Homework 4: Due Friday Sept 30@ 6: 00PM Ch4:#52,57,58,59,60,65,80,84,87,and90. Chapter 5: Rotation Angular variables Centripetal acceleration Kepler's laws Physics 121: Lecture 10, Pg 1

Physics 121: Lecture 10, Pg 1 Physics 121, Sections 9, 10, 11, and 12 Lecture 10 Today’s Topics: Homework 4: Due Friday Sept. 30 @ 6:00PM Ch.4: # 52, 57, 58, 59, 60, 65, 80, 84, 87, and 90. Chapter 5: Rotation Angular variables Centripetal acceleration Kepler’s laws

Chapter 5 Uniform circular Motion What does it mean How do we describe it What can we learn about it Physics 121: Lecture 10, Pg 2

Physics 121: Lecture 10, Pg 2 Chapter 5 Uniform Circular Motion What does it mean ? How do we describe it ? What can we learn about it ?

Review displacement, velocity, acceleration 3-D Kinematics: vector equations dr/dt a= d2r/dt2 th Velocity △r/△t dr/ dt △v Acceleration a dy/ at Physics 121: Lecture 10, Pg 3

Physics 121: Lecture 10, Pg 3 Review ( displacement, velocity, acceleration ) Velocity : Acceleration : a = dv / dt 3-D Kinematics : vector equations: r = r(t) v = dr / dt a = d2 r / dt2 v = dr / dt vav = r / t aav = v / t v2 -v1 v v2 y x path v1

General 3-D motion with non -zero acceleration path a au t a a a+0 because either or both change in magnitude of v atO change in direction of V 0 Uniform Circular Motion is one specific case of this Animation Physics 121: Lecture 10, Pg 4

Physics 121: Lecture 10, Pg 4 General 3-D motion with non-zero acceleration: Uniform Circular Motion is one specific case of this : a v path t a a a = 0 because either or both: -> change in magnitude of v -> change in direction of v a = 0 a = 0 a = a + a Animation

What is Uniform Circular Motion (UCM)? Motion in a circle with Constant Radius r a(x,y) Constant Speed v=v R 0 acceleration con Physics 121: Lecture 10, Pg 5

Physics 121: Lecture 10, Pg 5 What is Uniform Circular Motion (UCM) ? Motion in a circle with: Constant Radius R Constant Speed v = |v| acceleration ? R v x y (x,y) a = 0 a a = const

How can we describe UCM? n general, one coordinate system is as good as any other Cartesian: 》(X,y)[ position 》(vx,y) elocity] Polar (x,y) 》(R,0)[ position] 》(vR,o)[ velocity ∠0 R In UCM R is constant(hence VR=0) o(angular velocity)is constant Polar coordinates are a natural way to describe UCM! Physics 121: Lecture 10, Pg 6

Physics 121: Lecture 10, Pg 6 How can we describe UCM? In general, one coordinate system is as good as any other: Cartesian: » (x,y) [position] » (vx ,vy ) [velocity] Polar: » (R,) [position] » (vR ,) [velocity] In UCM: R is constant (hence vR = 0).  (angular velocity) is constant. Polar coordinates are a natural way to describe UCM! R v x y (x,y) 

Aside: Polar Unit vectors We are familiar with the Cartesian unit vectors: ijk Now introduce polar unit-vectors" r and e r points in radial direction e points in tangential(ccw) direction e Physics 121: Lecture 10, Pg 7

Physics 121: Lecture 10, Pg 7 Aside: Polar Unit Vectors We are familiar with the Cartesian unit vectors: i j k x y i j R  r ^  ^ ^ ^ ^ ^ Now introduce “polar unit-vectors” r and  : r points in radial direction  points in tangential (ccw) direction

Polar Coordinates: The arc length s(distance along the circumference)is related to the angle in a simple way s= R0, where 0 is the angular displacement units of e are called radians For one complete revolution 2TR= ROC )(x,y) R S 0 has period2π 1 revolution 2t radians Physics 121: Lecture 10, Pg 8

Physics 121: Lecture 10, Pg 8 Polar Coordinates: The arc length s (distance along the circumference) is related to the angle in a simple way: s = R, where  is the angular displacement. units of  are called radians. For one complete revolution: 2R = Rc c = 2  has period 2. R v x y (x,y) s  1 revolution = 2 radians

Polar coordinates X=R cos 0 y=R sin e R(X, y) COS sIn 0 0 π/2 3丌/22元 Physics 121: Lecture 10, Pg 9

Physics 121: Lecture 10, Pg 9 Polar Coordinates... ▪ x = R cos  ▪ y = R sin  R x y (x,y)      -1 1 0 cos sin 

Polar coordinates In Cartesian co-ordinates we say velocity AX/At=V vt In polar coordinates, angular velocity 40/At=o 0= ot o has units of radians/second Displacement s= vt R but s Re Rot so: OR Physics 121: Lecture 10, Pg 10

Physics 121: Lecture 10, Pg 10 Polar Coordinates... In Cartesian co-ordinates we say velocity x/ t = v. x = vt In polar coordinates, angular velocity / t = .  = t  has units of radians/second. Displacement s = vt. but s = R = Rt, so: R v x y s =t v = R