Help with finding normal modes of a bar swinging on a string

[BiotFartLaw]

Homework Statement

A bar with mass m and length l is attached at one end to a string (also length l) and is swinging back and forth. Find the normal modes of oscillation.

Homework Equations

L=T-U, and the Lagrange-euler equation, I=(1/12)ml^2

The Attempt at a Solution

So my idea is this. Use the two angles $\theta$ (angle of string from the vertical) and $\phi$ (angle of the bar from the vertical) as the generalized coordinates and set up a Lagrangian to get two (probably coupled) differential equations. THen I'm guessing two different eigenfrequencies will pop out when I solve them.

My problem is that I'm not sure what the (translational) kinetic energy is. (The rotational would just be I*ω^2 and this ω^2 would, in the end, give me the two frequencies [?]) I've tried finding $\dot{x}$ and $dot{y}$ using sines and cosines of the angles, but when I do (and I make appropriate approximations) I'm left with no $dot{\theta}$ or $dot{\phi}$terms which doesn't seem right.
But I'm also not sure if it's as simple as $T=ml^2(dot{\theta}+dot{phi})$.

Thanks

 

[vela]

The easiest thing to do is simply write down the x and y coordinates of the center of mass of the rod in terms of the two angles and then calculate $\frac{1}{2}(\dot{x}^2+\dot{y}^2)$.

 

[BiotFartLaw]

Thanks. A new question though:

What is ω (from the rotational KE) in terms of the two angles? Usually $\omega = dot{\theta}$. But that's only for one angle. Because if ω isn't in terms of the angle(s) then the rotational KE drops out when you take the derivatives for L which doesn't seem right.

Thanks again.

 

[vela]

From the way you defined the angles, you should have $\omega=\dot{\phi}$, right? Changing that angle corresponds to the bar rotating whereas changing $\theta$ only causes translation of the bar.

 

[paranormal]

for your question! Finding the normal modes of a bar swinging on a string can be a challenging problem, but I am happy to provide some guidance.

First, let's define our variables and coordinate system. We have a bar with mass m and length l, attached at one end to a string of length l. Let's use the angles θ and φ to represent the angles of the string and bar, respectively, from the vertical. We can also define the coordinates x and y as the horizontal and vertical displacements of the bar, respectively.

Next, let's consider the kinetic energy of the system. Since the bar is swinging, it will have both translational and rotational motion. The translational kinetic energy can be written as T = (1/2)m(dx/dt)^2 + (1/2)m(dy/dt)^2. We can express dx/dt and dy/dt in terms of θ and φ using trigonometric identities, but we also need to consider the rotational kinetic energy. Since the bar is rotating about its center of mass, we can use the parallel axis theorem to find the moment of inertia I = (1/12)ml^2 + m(l/2)^2 = (1/3)ml^2. The rotational kinetic energy can then be written as T_rot = (1/2)I(dθ/dt)^2 + (1/2)I(dφ/dt)^2.

Now, let's consider the potential energy of the system. The only potential energy present is due to gravity, which can be written as U = mgy. We can express y in terms of θ and φ using trigonometric identities.

Combining the kinetic and potential energies, we can write the Lagrangian as L = T - U. We can then use the Lagrange-Euler equations to find the equations of motion for θ and φ.

Solving these equations, we will find two natural frequencies ω1 and ω2 corresponding to the normal modes of the system. These frequencies can be found by setting the determinant of the coefficients of the differential equations to zero.

I hope this helps you in finding the normal modes of a bar swinging on a string. Remember to carefully consider the kinetic and potential energies of the system, and to use the Lagrange-Euler equations to find the equations of motion. Good luck with your homework!