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I am trying to solve question using the Euler–Lagrange equation and i am having trouble defining the potential energy.


Example: 1

A block of mass $m$ is held motionless on a frictionless plane of mass $M$ and angle of inclination $θ$. The plane rests on a frictionless horizontal surface. The block is released. What is the horizontal acceleration of the plane.

Here i have defined the coordinates:

  • Let $x_1$ be the horizontal coordinates of the plane (with positive $x_1$ to the left)
  • Let $x_2$ be the horizontal coordinates of the block (with positive $x_2$ to the right)

Diagram:

Blockquote

The relative distance between the plane and the block is $x_1 + x_2$

So the height fallen by the block is $(x_1 + x_2)tan⁡(θ)$

I have trouble working out which direction the potential energy will be positive, as we don't have a "$y$ coordinates" defined.


Example 2:

Two massless sticks of length $2r$, each with a mass $m$ fixed at its middles are hinged at an end. One stands on top of the other. The bottom end of the lower stick is hinged at the ground. They are held such that the lower stick is vertical, and the upper one is tilted at a small angle $ϵ$ with respect to the vertical. They are then released. At this instant, what are the angular accelerations of the two sticks? Work in the approximation where $ϵ$ is very small.

Blockquote

Coordinates: Let $θ_1 (t)$ and $θ_2 (t)$ be defined as shown.

  • The position of the bottom mass in cartesian coordinates is: $(r sin⁡(θ_1 ),r cos⁡(θ_1 ) )$
  • The position of the top mass is: $(2r sin⁡(θ_1 )-r sin⁡(θ_2 ),2r cos⁡(θ_1 )+r cos⁡(θ_2 ) )$

Thus the potential energy is $V(θ_1,θ_2 )=mgr(3 cos⁡(θ_1 )+cos⁡(θ_2 ))$

However again I have no idea whether the potential will be positive or negative.


What is the general method to find the direction of the positive potential energy?

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If you have to do positive work to move an object across a force field, then the potential energy is increasing in that direction. Normally, if you let go of the object, it will tend to move toward a position of lower potential energy. In a gravitational field, the P.E. increases as an object is moved “up” (maybe to a smaller negative value).

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