PLEASE ANSWER ALL PARTS A through D A particle of constant mass \"m\" is repelle

Question

PLEASE ANSWER ALL PARTS A through D

A particle of constant mass "m" is repelled from the origin by a position-dependent force F(x) that is inversely proportional to the cube of is distance from the origin:

F(x) = k / (x^3)

where "k" is a positive constant. The particle is at rest at a distance "x_0" (supposed to be x subscript 0) from the origin at t=0. (note that the last sentence proviedes two "inintial conditions" for this problem, so we should be able to solve this second-order differential equation.)

________________________________________________________________________________________

a) Integrate the equation of motion to find the velocity as a function of position, v(x). Be sure to use the initial conditions when integrating. (notice that integrating for velocity as a function of position yields the kinetic energy T=.5mv^2 as a function of position, T(x) )

b) integrate the equation for v(x) obtainted in part (a) with respect to time to show that the position as a function of time x(t) is given by:

Again, be sure to use the initial condition in your integration

c) find the potential energy function V(x) for F(x)

d) show that F(x) is a conservative force by showing that the value of the total mechanical energy as a function of position E(x) = T(x) + V(x) is the same for any position and hence, from (b), that "E" is a constant in time. (hint: recall the comment about "T" in part (a). Thus, with (c), you already have "T" and "V". Add them together to seee what you get.

Explanation / Answer

This is a 2-dimensional potential, and the resulting force will be a 2-dimensional vector with components: F(x,y) = (-dU/dx, -dU/dy) where the derivatives are partial derivatives. I suspect there is an error in the way you have written the potential function. Did you really mean: U(x,y) = 3*x^(3*y) - 7*x ?????? If so, then the force is given by: F(x,y) = (7 - 9*(y/x)*x^(3*y) ), -9*(ln(x))*x^(3*y)) If you meant something else, then simply take the partial derivatives of U(x,y) with respect to x and y. The x-component of the force is given by -1 times the derivative with respect to x, and the y-component is -1 times the derivative with respect to y

Related Questions

Navigate

• Browse (All)
• Browse P
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.