A particle of mass m moves under the influence of ...

Question

A particle of mass m moves under the influence of a potential energy large{U(x) = frac{a}{x} + bx}, where a and b are positive constants and the particle is restricted to the region x > 0. Find a point of equilibrium for the particle and demonstrate that it is stable.

Patrick Ford · Accepted Answer

In this problem, we'll use the relationship between equilibrium and potential energy to determine the equilibrium point.Force:[readmore]Fx=-∆Ux∆x Equilibrium condition:∆U∆x=0Power rule of derivation:ddt(xn)=nxn-1In equilibrium, the net force is zero.The derivative (slope) of the potential energy is the force. When the slope is zero, the mass is in equilibrium. This is the basis of the equilibrium condition.We'll derive U(x) and set it equal to zero, then solve for x.U(x) = ax-1 + bx dU(x)/dx = -ax-2 + b = 0ax-2 = bx-2 = b/ax2 = (a/b)x = sqrt(a/b) The point of equilibrium for the particle is sqrt(a/b).How to demonstrate stability:Stable:d2Udx2>0Unstable:d2U/dx2 < 0d2U(x)/dx2 = d/dx(dU(x)/dx) = d/dx(-ax-2 + b) = +2ax-3 + 0We have found that d2U(x)/dx2 > 0. Greater than zero implies that it should be positive. Hence, it is stable.

Problem: A particle of mass m moves under the influence of a potential energy large{U(x) = frac{a}{x} + bx}, where a and b are positive constants and the particle is restricted to the region x > 0. Find a point of equilibrium for the particle and demonstrate that it is stable.

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Problem: A particle of mass m moves under the influence of a potential energy large{U(x) = frac{a}{x} + bx}, where a and b are positive constants and the particle is restricted to the region x &gt; 0. Find a point of equilibrium for the particle and demonstrate that it is stable.

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Problem: A particle of mass m moves under the influence of a potential energy large{U(x) = frac{a}{x} + bx}, where a and b are positive constants and the particle is restricted to the region x > 0. Find a point of equilibrium for the particle and demonstrate that it is stable.