Classical Dynamics of Particles and Systems
Classical Dynamics of Particles and Systems
5th Edition
ISBN: 9780534408961
Author: Stephen T. Thornton, Jerry B. Marion
Publisher: Cengage Learning
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Chapter 4, Problem 4.1P
To determine

The potential corresponding to the motion of particle.

Expert Solution & Answer
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Answer to Problem 4.1P

The potential corresponding to the motion of particle isUxkdlx2+kld4l3x4.

Explanation of Solution

Consider the figure below.

Classical Dynamics of Particles and Systems, Chapter 4, Problem 4.1P

Unstretched length isl0. The mass m is stretched by a distance d. When the system moves a horizontal distance x, net force on the mass is,

    F=2ksl0sinθ

Here, F is the force on the mass.

Potential is given by,

    Ux=Fdx

Here, U is the potential.

From the diagram,

    sinθ=xs=xx2+l2

The unstretched length is given by,

    l0=ld

Replacesinθbyxx2+l2andl0byldin the expression for F.

    F=2ksl+dxx2+l2=2kx2+l2l+dxx2+l2=2kxx2+l2x2+l2l+d=2kx1ldl1+x2l21/2

Using Binomial theorem and expressing up to first two terms,

    F=2kx1ldl1x22l2=2kdlxkldl3x3

Conclusion:

Substitute for F in the expression for U and integrating,

    U=2kdlxkldl3x3dx=2kdlxdx+kld4l3x3dx=kdlx2+kld4l3x4

Therefore, potential corresponding to the motion of particle isUxkdlx2+kld4l3x4.

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