RuinaA2025DynSim – gyanantaran

problem01

Just one particle. 2D or 3D, your choice. Use a force, or forces that you like (gravity, spring, air friction). Any example of interest. Find a numerical solution. Graph it.…

problem02

Assume that the positions relative to an origin of four random points, which are randomly located in space are given as \(\vec{\mathbf{r}}_A\), \(\vec{\mathbf{r}}_B\), \(\vec…

problem03

Take a simple set of ODEs. Use a set you like,e.g., harmonic oscillator, non-linear pendulum, the Lorentz system (look it up on the internet). Solve this set numerically 3…

problem04

Note that, this distributive law implies that, for given \(va \times \vec{v}\) is a linear operator.That is \(va \times \vec{v}\) is a linear function fo \(\vec{v}\). Later…

problem05

I have carefully done some of the reading assigned. That which I don’t understand or agree with, I have discussed in class. I haven’t yet carefully read Chapter 3, Section…

problem06

One end of of a negligible-mass spring (k, L0) is pinned to the origin, the other to a mass (m). There is gravity (g). Initial Conditions (ICs): The initial position is \(\ve…

problem07

Draw a picture of some object (a face, a house, whatever), and make it move around on the screen in a smooth and interesting way. No distortions. Just motions and rotations

problem08

This is the simplest dynamics problem, but posed in polar coordinates. Assume a particle is on a plane with no force on it. So, you know it moves at constant speed in a…

problem09

A cannon ball m is launched at angle θ and speed v0. It is acted on by gravity g and a viscous drag with magnitude \(\|c\vec{v}\|\).

problem10

Consider a point mass hanging from a zero-restlength linear spring (`0 = 0) in a constant gravitational field.

problem11

These two problems are both about central forces. In both cases the only force is central (directed on the particle towards the origin) and only depends on radius: \(\vec{\ma…

problem12

A cannon ball \(m\) is launched at angle \(\theta\) and speed \(v_0\). It is acted on by gravity \(g\) and a quadratic drag with magnitude \(\|cv^2\|\).

problem13

2D. Equilibrium and dynamics of a point mass \(m\) at \(\vec{r}_{C}\). Gravity \(g\) points down in the minus \(\hat{j}\) direction. Points A and B are anchored at \(\vec{r}_…

problem14

Consider a moving particle P. Consider also a moving point C (moving relative to a Newtonian frame \(\mathcal{F}\) that has an origin 0). For which of these definitions of \(…

problem15

For each of the force fields below, find the associated potential energy function. All of them are 3D. In some cases the energy is associated with the positions of two…

problem16

16. Mechanics of two or more particles

For two unequal particles with mass \(m_1\) and \(m_2\) what is the period of circular motion if the distance between the…

problem17

(From Ruina/Pratap). Three equal masses, say \(m = 1\), are attracted by an inverse-square gravity law with \(G = 1\). That is, each mass is attracted to the other by \(F =…

problem18

The total kinetic energy of a system of particles is

problem19

You should master problem 14 before looking at this problem. Consider a system of moving particles with moving center of mass at G. Consider also a moving point C (moving…

problem20

This problem has 3 independent educational goals: