Hi again, I have a problem here withthis rod and collar exercise. I have solved

Question

Hi again, I have a problem here withthis rod and collar exercise. I have solved the first part of it,but I dont know how to solve the second part, because I cant solvefor and the radius of curvature for the rod. The equationof the rod is y = 1.5 - 1.5x^2, which I cant solvefor , I need this solve the normal force on the rod andthe rate of decrease of speed of the collar. I get 11.3 (usingmetric system formulae) for the speed when it reaches point B,which I think is right.
The 10 N collar has a speed of 1.5 metres/second at A. the attachedspring has an unstretched length of 0.6 metre and a stiffness of150 N/metre. If the collar moves over the smooth rod. Determine itsspeed when it reaches point B, the normal force of the rod on thecollar, and the rate of decrease in its speed. This problem is very similiar to onefound in Hibbeler, RC2007, Engineeringmechanics dynamics,Prentice Hall, Singapore, p.210. Any help would begreat. -nick.

Explanation / Answer

According to the law of conservation of energy we have K1 + U1 = K2 +U2 (1/2)mu2 + w * h = (1/2)mv2 + w *h1 w = m * g or m = (w/g) at point B we have y = 0 therefore we get y = 1.5 - 1.5x2 or 1.5 - 1.5x2 = 0 or x = 1 or 1.5 - 1.5x2 = 0 or x = 1 or (1/2) * (w/g) * u2 + w * h = (1/2) * (w/g) *v2 + w * h1 solving the above equation for v we get v2 = 2 * (u2 + gh -gh1) or v = [2 * (u2 + gh -gh1)]1/2 -----------(1) u = 1.5 m/s,g = 9.8 m/s,h = 1.5 m and h1 = 1m the normal force of the rod on the collar F = k * x + (-w) = k * x - w where k = 150 N/m,x = 0.6 m and w = 10 N rate of decrease in its speed is obtained by differentiatingthe equation of the rod with respect to time y = 1.5 - 1.5x2 (dy/dt) = -1.5 * 2 * (dx/dt) or vy = -3.0 * vx or vx = -(vy/3.0) vy is equal to the final speed of the rod obtainedin equation (1) and the negative value indicates the rate ofdecrease in its speed. or vx = -(vy/3.0) vy is equal to the final speed of the rod obtainedin equation (1) and the negative value indicates the rate ofdecrease in its speed.

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