Question

Assume without loss of generality that the parabola is described by , for A, B > 0, and that an object of mass m is situated initially at (x₀ , y₀ )= (0, A) at rest before being given a tiny nudge towards positive x.

y(1) = A - Bir

a) Use energy methods to determine the speed of the particle as a function of x.

b) Calculate the radius of curvature r(x) for the parabola.

c) Given dy/dx = tan($\Theta$), by definition, determine cos($\Theta$) in terms of dy/dx. (Draw an infinitesimal triangle.)

d) Use Newton II in the radial/normal direction to determine the magnitude of the normal force N(x) and show that it is always strictly positive for this problem.

Answer 1

use concepts of work and energy, trigonometry

U y = A - Bx2. (a) Applying work energy theore my (A-3) = 1 mcx]" ing (A - (A-B*2)) = Im. Cu(x) ? &(x) = √2g B x² . Radius of curvature = (tly r(x) = (1 + 48?x2337 2B 8(x) Tit 2B (c) ten o = -2BX dy dre coso = Vittanzo V1+ 489x2 Sea Scanned with . CamScanner