Using Cartesian Coordinates to Describe 2 D Motion
Using Cartesian Coordinates to Describe 2 D Motion Y P 1 y 1 v 1 R 1 y 2 P 2 R 2 j = (0, 1) motion path described by the particle on 2 D v 2 x 1 i = (1, 0) x 2 X R = x=ix(t) + y ij += y(t) (x, 0) (0, y)0)= +(x, (0, y)y(t)) = (x(t), y(t)) R(t) j =+(x(t), . . . . v(t) = R(t) = (x(t), y(t)) always in the tangent direction of the motion path a(t) = v(t) = (x(t), y(t))
Using Polar Coordinates to Describe 2 D Motion Y P 1 v 1 motion path described by the particle on 2 D R 1 eq eq ’ er r 1 q 1 er ’ R 2 v 2 X radial acceleration angle acceleration R(t) = r(t) er(t) q information is already contained in er, so eq does not show up in R(t) If R 1(t) = r 1(t) er, R 2(t) = r 2(t) er’, R 1(t) + R 2(t) (r 1(t) + r 2(t)) er since er(t) er’ (t). So, how do we do vector algebra in polar coordinates? . . . v(t) = R (t) = r(t) er(t) + r(t) er(t) = vr(t) er(t) + vq(t) eq(t) always in the tangent direction of the motion path. . a(t) = v (t) = r(t) er(t) + 2 r(t) er(t) + r(t) er(t) = ar(t) er(t) + aq(t) eq(t)
Using Tangential/Normal Coordinates to Describe 2 D Motion Using T/N coordinate, one is only interested in the velocity and acceleration of the Y moving particles. Hence, there is no position vector. P 1 motion path described It is best to imagine you are traveling along the by the 1 particle on 2 D motion path (see figure). T/N coordinate system relies on the arc-length variable, S, to measure the velocity and acceleration. Given any instant, say t 0, for any t > t 0 P To define the speed, consider t > t 0, the limit • It is no longer S(t) = limt t 0 S(t)/(t – t 0) required for a defines the velocity at point P 1. Imagine the limit as if there are fixed origin. t 3 > t 2 > t 1 > … > t 0 and measure the corresponding S(t 3), S(t 2), S(t 1), … In the limit t t 0, S(t) 0 and t – t 0 0. But its ratio measures the speed of the velocity. ) S(t 3 X ) S(t 2 t 1) S( S(t) measures the distance traveled by the particle From t 0 to t.
Using Tangential/Normal Coordinates to Describe 2 D Motion • But, S(t) measures the speed, which is the magnitude of the velocity. The direction of the velocity has not yet been specified. So, where is it? Recall from Cartesian, Polar coordinate systems, the velocity is always in the tangent direction of the motion path. P 1 Therefore, the velocity expressed in T/N coordinate is • v(t) = S(t) et (t) magnitude of velocity (tangential) direction of the velocity (see the red arrow) en et The acceleration follows immediately by differentiation: • • a(t) = v(t) = S(t) et(t) + S(t) et(t) = at(t) et(t) + an(t) en(t) tangent acceleration measures how fast S(t) changes in time. normal acceleration, which describes the change in the direction of motion and points toward the center of curvature at the given position of the motion path (see the blue arrow).
Using Tangential/Normal Coordinates to Describe 2 D Motion Y P 1 en et motion path described by the particle on 2 D en P 2 et X
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