Question

35. F=2x2-3x-2, then select the correct statement
fo
eg
1
D X =
is the position of stable equilibrium
2
127 x = 2 is the position of stable equilibrium
fo
-1
(3) X = = is the position of unstable equilibrium
2
(4) X = 2 is the position of neutral equilibrium​

Answer 1

Answer:

X = 1/2 = is the position of unstable equilibrium

Answer 2

Dear student,

It is given that force, F = 2x² - 3x - 2. We need to select the correct statement about its equilibrium position.

For a force to be in equilibrium, it has to be equal to zero. So,

            F = 2x² - 3x - 2

           2x² - 3x - 2 = 0

     2x² - 4x + x - 2 = 0

2x (x - 2) + 1( x - 2) = 0

(2x + 1) (x - 2) = 0

   x = - 1/2 or, x = 2

So, the possible positions for the positions x to be in equilibrium are  - 1/2 and 2.

Now, we know the following from the concepts of equilibrium and potential energy that;

For stable equilibrium

         Force, f = - [ dv / dr ]

                    f = - dv/ dr

         differentiating w.r.t. r;

               df/ dr = - d²v/ dr²

              df/ dr = - ve ( slope ) ____(1)

For unstable equilibrium

          Force, f = - [ - dv / dr ]

                    f = + dv/ dr

         differentiating w.r.t. r;

               df/ dr =  d²v/ dr²

               df/ dr = + ve ( slope ) ____(2)

With the help of data about the slopes, let us try to plot the position on the graph and conclude something about the slope. Refer to the attached image.

From the image it is evident that position - 1/2 represents negative slope and position + 2 represents positive slope. We know from (1) and (2) that they are for for stable and unstable equilibrium respectively.

Hence, option (1) x = - 1/2 is the position of stable equilibrium is the right answer.