Answer:
U = - (x⁴ / 4 - b x² / 2)
, c) The function is zero for x = 0 and √2
Explanation:
a and b) Strength and potential energy are related
F = - dU / dx
Therefore to find the energy we must integrate
∫ dU = -∫ F dx
∫ dU = - ∫ (a x³ –b x) dx
Let's make the integration
U = - (x⁴ / 4 - b x² / 2)
We evaluate the integral between the value
U - U₀ = -x⁴ / 4 + x² / 2 - (-x₀⁴ / 4 + x₀² / 2)
The arbitrary constant is zero, so that U is zero in the zero position
U₀ = 0 for x₀ = 0
c) Mechanical energy is the sum of kinetic energy plus potential energy
Em = K + U
Em = ½ m v² + ½ (x² -x⁴ / 2)
E = ½ m v² + ½ x² (1 - x² / 2)
Energy is positive
2 (E –K) = x² (1-x² / 2)
At the return points K = 0
The zero points of this function are
x = 0
(1- x² / 2) = 0
x² = 2
x = √ 2
The function is zero
x = 0 and √2
d) the movement is bounded for energy values less than or equal to
E <= ½ x² (1-x² / 2)
e) for this part we resolved Newton's second law
F = m a
ax³ - b x = m d²x / dt²
d²x / dt² = -b / m x + a / m x³3
The linear term gives a simple harmonic movement
w₀² = b / m
d²x / dt² = - w₀² x + a / m x³
The frequencies are the frequencies of the harmonic movement plus a small change due to the non-harmonic part of the movement
