Moment of inertia of rods

Ok so I’m extremely comfortable with calculating moment of inertia of continuous bodies but how do we do it for a system not continuous.
For example if 3 rods of mass $m$ and length $l$ are joined together to form an equilateral triangle what will be the moment of inertia about an axis passing through its centre of mass perpendicular to the plane.
i know that moment of inertia of each rod is $ml^2/12$ and c.o.m is at centroid?
also if 2 rods form a cross then to calculate the moment of inertia about its point of intersection would it be correct to sum up the individual moment of inertia of the rods form??

Robot speeds in body frame

I am building a robot with two wheels (and differential drive) and I am trying to make it have the same performances over very different loads (an order of magnitude between the ), so I decided to try to identify the carried mass.
I then wrote a model to be used in an observer and, since I have an Inertial Measurement Unit which outputs angular speeds and accelerations in the body frame (and the roll and pitch angles), I’ve decided to use the longitudinal speed $dot{x}$ and the angular speed $omega$ as states of my model. I’m not interested in the absolute position of the robot (it will be radiocontrolled), so I guess a 2nd order model would suffice.
The origin lies in the middle of the imaginary link between the centers of the two wheels and, since the position of the center of mass is not known and could vary (altough it is supposed to vary only when the robot is stopped), I’ll call $overrightarrow{G} = begin{bmatrix}G_x & G_y & G_zend{bmatrix}^T$ the vector which describes its position.

This is the model I wrote using the rigid body equations for planar motions (using the point $ begin{bmatrix} 0 & G_y & 0 end{bmatrix}^T $ to write the second equation):

$$begin{align}
ddot{x} &= frac{1}{m}sum F_i – begin{bmatrix}omega^2 & dot{omega} & 0end{bmatrix}overrightarrow{G}\
dot{omega} &= frac{1}{J}sum T_i – m(mathbb{R}-G_y)G_xomega^2
end{align}$$

Notes:

  1. $sum T_i$ includes the force $mddot{x}$
  2. Turn radius of center of mass $|mathbb{R}^*| = sqrt{G_x^2 + (mathbb{R}-G_y)^2}$

But I feel like something is wrong and I can’t understand why:

  1. The influence of $dot{omega}_z$ on $ddot{x}$ does not depend on the turn radius $mathbb{R}$ of the axle.
  2. The signs of the coupling terms maybe are wrong.

Any help which will cancel the fog of these doubts (or point out the errors I have made) would be great.

Frequency of Oscillations about Circular Orbit [closed]

I’m trying to figure out the frequency of small oscillations about the basic circular path of a mass at the end of a spring, being spun around a table. I understand that the spring will stretch out a bit, and then it will have a stable circular orbit. Moreover, introducing a little bit of extra stretching will cause it to oscillate about this orbit, but I have no idea what approach to take. I’ve seen things like binomial expansions and the like to calculate small changes to orbits, but I can’t even seems to get a good, functional equation to describe the set up. Can someone give me a little guidance?

Where does de Broglie wavelength $lambda=h/p$ for massive particles come from?

I’m curious where the expression $p=frac{h}{lambda}$ comes from. I know that for light, the following is true:

$E=pc$ and $E=hf$

so,

$pc=hf Rightarrow p=frac{hf}{c}=frac{h}{lambda}$

But how is the expression obtained for say, an electron where $Eneq pc$? I’ve read some people claim that the expression can be derived, and others saying it’s an experimentally verified relationship.

Can we prove Conservation of Angular Momentum without assuming internal forces are central?

The only proof I’ve seen of Conservation of Angular Momentum assumes that the internal forces of the system act along the line joining particles i and j (i.e. – all internal forces are central.) Can we prove Conservation of Angular Momentum without this assumption?

If not, how can we apply conservation of angular momentum to systems that involve collisions, such as the following:

A uniform circular turntable is at rest in the xy plane and is mounted on a frictionless axle, which lies along the vertical z axis. I throw a lump of putty (mass m) with speed v toward the edge of the turntable. When the putty hits the turntable, it sticks to the edge, and the two rotate together with angular velocity w. Find w.