Question

The potential energy of a particle in a certain field has the form U=(a/r^2) - (b/r) , where a and b are positive constants.r is the distance from center of the field.find the value of r0 corresponding to the equilibrium position of particle and examine whether this position is steady.

Answer 1

See the above image...

Answer 2

At equilibrium, the particle will try to have minimum potential energy.

To find the minimum of any function f with respect to x, we need to put df/dx = 0.

Here, we need to find the minimum of the potential energy function U = a/r2 - b/r
So, we have to differentail U with respect to r,

and then put dU/dr = 0 to find out the position where U is minimum.

U = ar-2 -br-1

dU/dr = -2ar-3 - (-br-2) = (-1/r2)(2a/r - b)

Putting dU/dr = 0,

(-1/r2)(2a/r - b) = 0

or, 2 a/r - b = 0, because 1/r2 = 0 will give r = infinity, which is anyway true

Hence, r = b/2a is the equilibrium distance