To determine the angular frequency of the oscillation of a particle under the given potential energy function, we first need to find the effective force acting on the particle by taking the negative gradient of the potential energy:
F(x)=−‌Given the potential energy function:
U(x)=−‌ we can compute the force as follows:
F(x)=−‌(−‌)Performing the differentiation using the quotient rule for
‌, where
u=αx and
v=x2+β2 ‌(‌)=‌| (α)(x2+β2)−(αx)(2x) |
| (x2+β2)2 |
Simplifying the numerator:
αx2+αβ2−2αx2=αβ2−αx2 So the force is:
F(x)=−‌In simple harmonic motion, near the equilibrium position at
x=0, the potential energy function can be approximated as a quadratic function of
x. That is:
U(x)≈‌kx2 where
k is the effective spring constant. The force near the equilibrium can also be approximated as:
F(x)≈−kx Here, we can determine
k by comparing the second derivative of the potential energy function at
x=0.
Taking the second derivative of
U(x) with respect to
x :
‌=‌(‌)We previously found the first derivative as:
‌=‌Evaluating the second derivative at
x=0 :
‌|x=0=‌(‌)|x=0Since the potential energy is symmetric around
x=0 and the first derivative at
x=0 is zero, we get:
‌|x=0=‌(‌)|x=0So, the effective spring constant
k is:
k=‌ The angular frequency
ω of the oscillation for a mass
m attached to a spring constant
k is given by:
ω=√‌Substituting the value of
k :
ω=√‌This shows that the angular frequency is proportional to:
√‌Thus, the correct option is:
Option C:
√‌