A bead of mass m is constrained to move on a frictionless wire. The wire lies in

Question

A bead of mass m is constrained to move on a frictionless wire. The wire lies in the x-y plane, and the height y of the wire as a function of horizontal position x is given by y = Ax. As (i) the position of the bead is completely determined by the single coordinate x and (ii) the system is conservative, we can use the conservation of energy to find the bead's position x(t). a) Find the specific equation for this system (written in terms of the dynamical variable x) that is equivalent to the general expression (4.136) for the total energy E. b) Then use separation of variables to solve this equation for x(t), assuming the initial conditions x(0) = 0 and upsilon_x(0) = upsilon_0. Express your answer for x(t) in terms m, g, upsilon_0, and A. c) Discuss the result in (b) in the limits A rightarrow 0 and A rightarrow infinity. Do these limits make sense? Explain. d) Starting with Newton's second law, find the equation of motion for the coordinates x. e) Calculate the time derivative of the equation you wrote down in(a) and confirm the equation of motion found in(d) guarantees energy is conserved.

Explanation / Answer

y = Ax

kinetic energy of the bead   K = mv2/2

Potential energy U = mgy   , y is the height

Total energy   E = m/2 (dx/dt)2 + mgy    , E is constant, conservation of energy

re-arraning the terms and replace y = Ax

dx/dt = (2E/m -2gAx)1/2

initial velocity v(0) = vo

vo2 = 2E/m

dx/(vo2 -2gAx)1/2   = dt

integrating both sides

(-1/gA)(vo2 -2gAx)1/2 = t + C   , C is an integration constant

inital condition t=0 x =0

C = -vo/gA

re-arranging the terms

x(t) = -gAt2 /2+vot

as A --> 0   x(t) = vot ,

the object moves on horizontal plane (y=0) at constant velocity vo

as A --> infinity x --> -infinity   , the inclined plane tends to become verticle

x increase until vo >= gAt and then decreases.

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