Solve Linked Masspoints Equation of Motion: α,l,m,g

[carllacan]

Homework Statement

Two particles of equal mass, one restricted to move along the y-axis and one restricted to move along the x axis, are linked by a solid rod of length l. Obtain the Lagrangian for the generalized coordinate α, defined as the angle of the rod with the horitzontal (see picture) and solve the equations of motion.
https://www.dropbox.com/s/h43s1749z7852oh/2014-01-30 22.50.54.jpg

Homework Equations

The Attempt at a Solution

My lagrangian looks like this: L = (m/2)l²$\dot{α}$² -mglsin(α)

From it I obtain $\ddot{α}$ = (g/l) cos(α)

But I don't know how to solve that. I know how to use the small angle approximation for the case when there is a sin. Here I am clueless.

And I am algo given a hint which confounds me: I am told to solve for t(α) applying the following equation: https://www.dropbox.com/s/hl2xfm07wt40kdn/2014-01-30 23.00.23.jpg

 

[ehild]

Let $\omega=\dot \alpha$.

$$\ddot \alpha = \frac{d \omega}{d \alpha} \frac {d \alpha }{dt}=0.5\frac{d(\omega^2)}{d \alpha} =\frac {g}{l}\cos(\alpha)$$

Integrate. You get ω=dα/dt as function of alpha. Integrate again.

ehild

 

[carllacan]

Thank you, that was a clever trick! Now i can get to the equation mentioned on the hints. However I cannot go on, as the integral seems rather complex (i solved it with wolfram and ran out of computing time).

The hint actually tells me to give the result as t(α) applying the function F there defined. Would you interpret that as that I can just give t(α) = (some constants)*F(b, α)? (Yes, I know I should just ask the professor, but I have a few problems with that)

 

[ehild]

The hint actually tells me to give the result as t(α) applying the function F there defined. Would you interpret that as that I can just give t(α) = (some constants)*F(b, α)? (Yes, I know I should just ask the professor, but I have a few problems with that)

I would say you can use the hint given.


ehild

 

[paranormal]

I would approach this problem by first understanding the physical setup and the variables involved. From the given picture and information, it appears that we have two particles of equal mass connected by a rod of length l, with one particle restricted to move along the x-axis and the other along the y-axis. The angle α is defined as the angle of the rod with the horizontal axis.

Next, I would use the Lagrangian approach to solve for the equations of motion. The Lagrangian is a function that describes the dynamics of a system in terms of generalized coordinates, which in this case is the angle α. The Lagrangian is given by L = T - V, where T is the kinetic energy and V is the potential energy.

In this case, the kinetic energy is given by the sum of the kinetic energies of the two particles, which can be expressed as (m/2)(l^2)(α̇^2), where m is the mass of the particles and α̇ is the time derivative of α. The potential energy is given by the gravitational potential energy, which is -mgl sin(α). Therefore, the Lagrangian for this system is L = (m/2)(l^2)(α̇^2) - mgl sin(α).

Using the Euler-Lagrange equation, we can obtain the equation of motion for α, which is given by d/dt(dL/dα̇) - (dL/dα) = 0. This results in the equation \ddot{α} = (g/l) cos(α).

To solve this equation, we can use the given hint of solving for t(α) using the equation provided. This equation is known as the elliptic integral of the first kind and can be solved using numerical methods. Alternatively, we can also use the small angle approximation for small values of α, which simplifies the equation to \ddot{α} = (g/l)α. This is a simple harmonic oscillator equation and can be solved analytically.

In conclusion, the Lagrangian approach allows us to obtain the equations of motion for a system in terms of generalized coordinates. In this case, we obtained the equation of motion for the angle α and can solve it using numerical methods or the small angle approximation.