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Jenny Physics

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- Homework Statement
- A mass ##m## slides on a curve described by the equation ##y=f(x)## under the action of gravity (neglect any other forces). Write the equation of motion in the general case and in the case of small angle between the tangent to the curve and the horizontal (i.e. when ##dy/dx## is small).

- Relevant Equations
- One way is to use conservation of energy ##T=\frac{1}{2}m(\dot{x}^{2}+\dot{y}^{2}),U=-mgy## so ##T+U=\frac{1}{2}m(\dot{x}^{2}+\dot{y}^{2})-mgy=constant##.

So ##T+U=\frac{1}{2}m(\dot{x}^{2}+\dot{y}^{2})-mgy=constant##. If I derive this with respect to ##t##

$$\dot{x}\ddot{x}+\dot{y}\ddot{y}-g\dot{y}=0$$

Then I use ##\dot{y}=\dot{x}\frac{dy}{dx},\ddot{y}=\ddot{x}\frac{dy}{dx}+\dot{x}^{2}\frac{d^{2}y}{dx^{2}}##

to get

$$\ddot{x}+\frac{dy}{dx}\left[\ddot{x}\frac{dy}{dx}+\dot{x}^{2}\frac{d^{2}y}{dx^{2}}\right]-g\frac{dy}{dx}=0$$

This is the general equation. From here not sure how to simplify in the small limit without making several assumptions.

$$\dot{x}\ddot{x}+\dot{y}\ddot{y}-g\dot{y}=0$$

Then I use ##\dot{y}=\dot{x}\frac{dy}{dx},\ddot{y}=\ddot{x}\frac{dy}{dx}+\dot{x}^{2}\frac{d^{2}y}{dx^{2}}##

to get

$$\ddot{x}+\frac{dy}{dx}\left[\ddot{x}\frac{dy}{dx}+\dot{x}^{2}\frac{d^{2}y}{dx^{2}}\right]-g\frac{dy}{dx}=0$$

This is the general equation. From here not sure how to simplify in the small limit without making several assumptions.

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