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: 2R = Rc 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 = Rt, so: R v x y s =t v = R