Deriving an equation of motion

In summary, a uniform thin rod of length L and mass m is pivoted at one end and attached to the top of a car accelerating at a rate A. The equilibrium angle between the rod and the top of the car can be found using Newton's 2nd law for non-inertial frames, which gives tan(equilibrium angle) = g/A. To derive the equation of motion for a small angle phi from the equilibrium, we can calculate the torque acting on the rod and use the equation I * alpha = torque, where I is the moment of inertia and alpha is the angular acceleration. By replacing theta with phi + tan^{-1}(g/A), we can simplify the equation for small phi. It is stable at the
  • #1
matpo39
43
0
a uniform thin rod of length L and mass m is pivoted at one end the point is attached to the top of a car accelerating at a rate A.

a) what is the equilibrium angle between the rod and the top of the car?
b) suppose that the rod is displaced a small angle phi from the equilibrium derive the equation of motion for phi. Is the equilibrium angle stable or unstable?

I was able to get part a which using Newtons 2nd law for non inertial frames is
tan(equilibrium angle) = g/A.

but i am stuck on part b. I could use the lagrangian method to get phi(double dot) but that would be really messy, and was wondering if there would be a better way of getting it. oh and also he gave the hint to ignore air resistance and that torque= I*alpha.

thanks
 
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  • #2
matpo39 said:
a uniform thin rod of length L and mass m is pivoted at one end the point is attached to the top of a car accelerating at a rate A.

a) what is the equilibrium angle between the rod and the top of the car?
b) suppose that the rod is displaced a small angle phi from the equilibrium derive the equation of motion for phi. Is the equilibrium angle stable or unstable?

I was able to get part a which using Newtons 2nd law for non inertial frames is
tan(equilibrium angle) = g/A.

but i am stuck on part b. I could use the lagrangian method to get phi(double dot) but that would be really messy, and was wondering if there would be a better way of getting it. oh and also he gave the hint to ignore air resistance and that torque= I*alpha.

thanks

Getting the equation isn't hard but solving it might be...

I'm going to take theta to be the total angle from the horizontal, and phi as the angle from the equilibrium point.

Calculate the torque acting on the rod... this is in terms of theta. Then write down I * alpha = torque.

Plug in the value of I in terms of m and L. Plug in [tex]\alpha=\frac{d^2\theta}{dt^2}[/tex]

Then finally replace [tex]\theta[/tex] with [tex]\phi + tan^{-1}(g/A)[/tex]

That seems to give you the equation you need. It's a second order diff. equation... I'm not sure of the solution right now. But do you need the solution?
 
  • #3
thanks for the help, it seems i forgot that alpha= phi(double dot), as soon i understood that the rest of the problem wasnt too hard.
 
  • #4
The differential equation may be simplified for small [tex]\phi[/tex], since

[tex]\sin\phi = \phi[/tex] and

[tex]\cos\phi = 1[/tex] (approximately).

I don't know how to write ~ in latex.
 
  • #5
matpo39 said:
thanks for the help, it seems i forgot that alpha= phi(double dot), as soon i understood that the rest of the problem wasnt too hard.

I'm curious about the final solution to this problem... Did you have to solve the diff. eq? If you did, can you post it?

It seems stable to me because the second derivative of phi, is zero at the equilibrium point... but I'm not sure if this is the correct reasoning.
 
  • #6
This what you're looking for?

[tex]\approx[/tex]
 
  • #7
Thank you Hurkyl, that's what I was looking for.
 

1. What is an equation of motion?

An equation of motion is a mathematical representation of the relationship between an object's position, velocity, and acceleration over time. It describes how an object's motion changes in response to external forces.

2. How is an equation of motion derived?

An equation of motion can be derived using principles from classical mechanics, such as Newton's laws of motion. It involves analyzing the forces acting on an object and using calculus to determine the object's position, velocity, and acceleration at any given time.

3. What are the key variables in an equation of motion?

The key variables in an equation of motion are time, position, velocity, and acceleration. Time is typically denoted as "t", position as "x", velocity as "v", and acceleration as "a". These variables can be either scalar or vector quantities.

4. What are the different types of equations of motion?

There are three main types of equations of motion: uniform motion, uniformly accelerated motion, and non-uniformly accelerated motion. Uniform motion occurs when an object moves at a constant velocity, uniformly accelerated motion occurs when an object's acceleration is constant, and non-uniformly accelerated motion occurs when an object's acceleration changes over time.

5. How are equations of motion used in real-world applications?

Equations of motion are used in many different fields, including physics, engineering, and astronomy. They can be used to predict the motion of objects and design systems, such as rockets and satellites. They are also used in sports and video game development to create realistic movements.

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