When a particle is a distance r from the origin, its potential energy function is given by the equation U(r)=kr, where k is a constant and r=x2+y2+z2 (a) What are the SI units of k? (b) Find a mathematical expression in terms of x, y, and z for the y component of the force on the particle. (c) If U=3.00 J when the particle is 2.00 m from the origin, find the numerical value of the y component of the force on this particle when it is at the point (-1.00 m, 2.00 m, 3.00 m).

Question

When a particle is a distance r from the origin, its potential energy function is given by the equation U(r)=kr, where k is a constant and r=x2+y2+z2  (a) What are the SI units of k?  (b) Find a mathematical expression in terms of x, y, and z for the y component of the force on the particle.    (c) If U=3.00 J when the particle is 2.00 m from the origin, find the numerical value of the y component of the force on this particle when it is at the point (-1.00 m, 2.00 m, 3.00 m).

Jeffrey Jordon · Accepted Answer

First convert to cartesian U(x,y,z)=kr=k(x2+y2+z2)1/2 Now, in vague terms, the force is sort of, the &quot;rate of change&quot; of the potential energy function. E.g. gravity. The stronger the force, the more work you have to do to move upwards against it, and the more potential energy you gain as a result. So, from this, we can skip a bunch of math and take the leap to saying that the force in the Y direction depends only upon how U changes *in the y direction*. &quot;How U changes in the y direction&quot; is basically what the partial derivative of U with respect to y tells you. In general, I need four dimensions to plot U(x,y,z). However, in that 4D space, if I choose to move only in the y direction (*keeping x and z constant*), the rate of change of U that I measure *along that line* is the partial derivative of U with respect to y. Mathematically: Fy=U(x,y,z)y=yk(x2+y2+z2)1/2  = the derivative of U(x,y,z) with respect to y, *assuming that x and z are constants* This is not the total rate of change of U, but only the rate of change you&#039;d measure if you confined yourself to movement in the y direction. Hence, it&#039;s a partial derivative.

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