First calculate all the displacements x and y as below using right angled triangle theorems
x l cos
y l sin
Where l is the length of the bar
Next calculate the potential energy of the spring
1
P.Espring ky 2
2
Substitute the values in the above equation
1
2
P.Espring k l sin
2
Next find the potential energy of the force P
P.E 2 Px
Substitute the value in the above equation
P.E 2 Pl cos
Hence total potential energy of the system is given below
1
2
V k l sin 2 Pl cos
2
Hence differentiate the above equation
dV
kl 2 sin cos 2 Pl sin
d
kl 2
sin 2 2 Pl sin
2
d 2V
0
0 d 2
For stability at
,
. Hence differentiate the above equation and substitute
0
d 2V
kl 2 cos 2 2 Pl cos
2
d
0
Substitute
in the above equation
2
dV
kl 2 2 Pl 0
2
d
Hence
P
kl
2
Hence the range of values of P is shown below
kl
P
2