Dynamics Snapshot

Relevant wiki: Lagrangian Mechanics

Here's a Lagrangian mechanics solution:

Spatial Coordinates:

$x = x \\ y = x^2 \\ \dot{x} = \dot{x} \\ \dot{y} = 2 x \dot{x}$

Square of velocity and kinetic energy:

$v^2 = \dot{x}^2 + \dot{y}^2 = (1 + 4x^2) \dot{x}^2 \\ E = \frac{1}{2} m v^2 = \frac{1}{2} m (1 + 4x^2) \dot{x}^2$

Gravitational potential energy:

$U_g = m g y = m g x^2$

Spring potential energy:

$U_s = \frac{1}{2} k (x^2 + x^4)$

Lagrangian:

$L = E - U_g - U_s \\ = \frac{1}{2} m (1 + 4x^2) \dot{x}^2 - m g x^2 - \frac{1}{2} k (x^2 + x^4)$

Equation of Motion:

$\frac{d}{dt} \frac{\partial{L}}{\partial{\dot{x}}} = \frac{\partial{L}}{\partial{x}}$

Inner left term:

$\frac{\partial{L}}{\partial{\dot{x}}} = m (1 + 4x^2) \dot{x}$

Left term:

$\frac{d}{dt} \frac{\partial{L}}{\partial{\dot{x}}} = m (1 + 4x^2) \ddot{x} + 8 m x \dot{x}^2$

Right term:

$\frac{\partial{L}}{\partial{x}} = 4 m x \dot{x}^2 - 2 m g x - k x - 2 k x^3$

Equating:

$m (1 + 4x^2) \ddot{x} + 8 m x \dot{x}^2 = 4 m x \dot{x}^2 - 2 m g x - k x - 2 k x^3 \\ m (1 + 4x^2) \ddot{x} = -4 m x \dot{x}^2 - 2 m g x - k x - 2 k x^3$

Final expression for horizontal acceleration:

$\ddot{x} = \frac{-4 m x \dot{x}^2 - 2 m g x - k x - 2 k x^3}{m (1 + 4x^2)}$

Plugging in numbers:

$\ddot{x} = \frac{-4 (1) (\frac{1}{2}) (3^2) - 2 (1) (10) (\frac{1}{2}) - (5) (\frac{1}{2}) - (2)(5) (\frac{1}{2})^3}{(1) (1 + 4(\frac{1}{2})^2)} = \boxed{-15.875}$