Equation of motion of a mass on a 2d curve

In summary, the conversation discusses the equation of motion for a mass sliding on a curve under the action of gravity. The equation is derived using the conservation of energy and then simplified using other assumptions. There is a sign error in the original equation and the simplified equation also has a missing factor of 1/2. The final simplified equation shows what happens when the slope of the curve is small compared to 1.
  • #1
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.
 
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  • #2
When dy/dx is very small compared to 1 then its square will be negligible in that equation.
While not directly given by the small dy/dx the problem statement might also assume that d2y/dx2 is small.
 
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  • #3
What is wrong with $$\left(\frac{dx}{dt}\right)^2\left[1+\left(\frac{dy}{dx}\right)^2\right]-gy(x)=C$$ or $$\frac{dx}{dt}=\sqrt{\frac{C+gy}{\left[1+\left(\frac{dy}{dx}\right)^2\right]}}$$
 
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  • #4
Jenny Physics said:
Problem 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##.

You have a sign error: if [itex]y[/itex] increases then the potential energy should also increase.

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

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

Easier to use [itex]\dot y = f'(x) \dot x[/itex] before differentiating with respect to [itex]t[/itex]. Then [itex]\ddot y[/itex] doesn't appear.

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.

Chestermiller said:
What is wrong with $$\left(\frac{dx}{dt}\right)^2\left[1+\left(\frac{dy}{dx}\right)^2\right]-gy(x)=C$$ or $$\frac{dx}{dt}=\sqrt{\frac{C+gy}{\left[1+\left(\frac{dy}{dx}\right)^2\right]}}$$

I think you are missing a factor of 1/2 from the KE, and again the PE term has the wrong sign. But this does show most clearly what happens when [itex]|f'(x)| \ll 1[/itex].
 
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  • #5
pasmith said:
I think you are missing a factor of 1/2 from the KE, and again the PE term has the wrong sign. But this does show most clearly what happens when [itex]|f'(x)| \ll 1[/itex].
Yes, you're right. I left out a factor of 2. It should read: $$\frac{dx}{dt}=\sqrt{\frac{C+2gy}{\left[1+\left(\frac{dy}{dx}\right)^2\right]}}$$The sign of the PE term is correct if y is measured downwards. And, of course, it is trivial to get the equation when f' is negligible; I didn't feel I needed to add that.
 
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Related to Equation of motion of a mass on a 2d curve

What is the equation of motion of a mass on a 2d curve?

The equation of motion of a mass on a 2d curve is a mathematical representation of the position, velocity, and acceleration of a mass as it moves along a curved path in a two-dimensional space. It is commonly expressed as a parametric equation in terms of time.

What factors affect the equation of motion of a mass on a 2d curve?

The equation of motion of a mass on a 2d curve is affected by various factors such as the shape of the curve, the mass of the object, the initial velocity and position, and the forces acting on the mass. Other factors such as friction, air resistance, and gravity may also play a role in the equation of motion.

How is the equation of motion of a mass on a 2d curve derived?

The equation of motion of a mass on a 2d curve is derived using principles of calculus and Newton's laws of motion. It involves breaking down the motion into small intervals, determining the forces acting on the mass at each interval, and then using equations of motion to calculate the position, velocity, and acceleration of the mass at each point along the curve.

Can the equation of motion of a mass on a 2d curve be used to predict future motion?

Yes, the equation of motion of a mass on a 2d curve can be used to predict the future motion of an object. By plugging in different values for time, the equation can calculate the position, velocity, and acceleration of the mass at any given point in time, allowing for predictions of its future path.

What are some real-world applications of the equation of motion of a mass on a 2d curve?

The equation of motion of a mass on a 2d curve has numerous applications in fields such as physics, engineering, and astronomy. It is used to study the motion of objects in orbit, the trajectory of projectiles, and the movement of particles in fluid dynamics. It also plays a crucial role in designing roller coasters and other amusement park rides.

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