This week's sets of classical pen and paper and computational exercises deal with simple motion problems and conservation laws; energy, momentum and angular momentum. These conservation laws are central in Physics and understanding them properly lays the foundation for understanding and analyzing more complicated physics problems.
The relevant reading background is
In both textbooks there are many nice worked out examples. Malthe-Sørenssen's text contains also several coding examples you may find useful.
The numerical homework focuses on another motion problem where you can use the code you developed in homework 3, almost entirely. Please take a look at the posted solution (jupyter-notebook) for homework 3. You need only to change the forces at play. The numerical problem this time is based on your code from homework 3 and we will try to make the motion of a falling object in two dimensions more realistic by allowing to bounce up again due to a normal force from the floor.
Consider a particle of mass \( m \) moving according to the potential
$$ V(x,y,z)=A\exp\left\{-\frac{x^2+z^2}{2a^2}\right\}. $$At \( t=0 \) we have a single object with position \( \boldsymbol{r}_0=x_0\boldsymbol{e}_x+y_0\boldsymbol{e}_y \).We add also a force in the \( x \)-direction at \( t=0 \). We assume that the object is at rest at \( t=0 \).
$$ \boldsymbol{F} = F\boldsymbol{e}_x. $$A particle of mass \( m \) has velocity \( v=\alpha/x \), where \( x \) is its displacement.
A particle is thereafter under the influence of a force \( F=-kx+kx^3/\alpha^2 \), where \( k \) and \( \alpha \) are constants and \( k \) is positive.
This exercise builds on the code you wrote for solving homework 3. We recommend strongly that you study the text of Malthe-Sørenssen, section 7.5.
In homework 3 we introduced gravity and air resistance and studied their effects via a constant acceleration due to gravity and the force arising from air resistance. But what happens when the ball hits the floor? What if we would like to simulate the normal force from the floor acting on the ball? This exercise shows how we can include more complicated forces with no pain! And the force we include here is an example of a case where analytical solutions may either be difficult to find or we cannot find an analytical solution at all.
We need then to include a force model for the normal force from the floor on the ball. The simplest approach to such a system is to introduce a contact force model represented by a spring model. We model the interaction between the floor and the ball as a single spring. But the normal force is zero when there is no contact. Here we define a simple model that allows us to include such effects in our models.
The normal force from the floor on the ball is represented by a spring force. This is a strong simplification of the actual deformation process occurring at the contact between the ball and the floor due to the deformation of both the ball and the floor.
The deformed region corresponds roughly to the region of overlap between the ball and the floor. The depth of this region is \( \Delta y = R − y(t) \), where \( R \) is the radius of the ball. This is supposed to represent the compression of the spring. Our model for the normal force acting on the ball is then
$$ \boldsymbol{N} = −k (R − y(t)) \boldsymbol{e}_y. $$The normal force must act upward when \( y < R \), hence the sign must be negative. However, we must also ensure that the normal force only acts when the ball is in contact with the floor, otherwise the normal force is zero. The full formation of the normal force is therefore
$$ \boldsymbol{N} = −k (R − y(t)) \boldsymbol{e}_y, $$when \( y(t) < R \) and zero when \( y(t) \le R \). In the numerical calculations you can choose \( R=0.1 \) m and the spring constant \( k=1000 \) N/m.
The following gives you an opportunity to earn five extra credit points on each of the remaining homeworks and ten extra credit points on the midterms and finals. This assignment also covers an aspect of the scientific process that is not taught in most undergraduate programs: scientific writing. Writing scientific reports is how scientist communicate their results to the rest of the field. Knowing how to assemble a well written scientific report will greatly benefit you in you upper level classes, in graduate school, and in the work place.
The full information on extra credits is found at https://github.com/mhjensen/Physics321/blob/master/doc/Homeworks/ExtraCredits/. There you will also find examples on how to write a scientific article. Below you can also find a description on how to gain extra credits by attending scientific talks.
This assignment allows you to gain extra credit points by practicing your scientific writing. For each of the remaining homeworks you can submit the specified section of a scientific report (written about the numerical aspect of the homework) for five extra credit points on the assignment. For the two midterms and the final, submitting a full scientific report covering the numerical analysis problem will be worth ten extra points. For credit the grader must be able to tell that you put effort into the assignment (i.e. well written, well formatted, etc.). If you are unfamiliar with writing scientific reports, see the information here
The following table explains what aspect of a scientific report is due with which homework. You can submit the assignment in any format you like, in the same document as your homework, or in a different one. Remember to cite any external references you use and include a reference list. There are no length requirements, but make sure what you turn in is complete and through. If you have any questions, please contact Julie Butler at butler@frib.msu.edu.
| HW/Project | Due Date | Extra Credit Assignment |
|---|---|---|
| HW 3 | 2-8 | Abstract |
| HW 4 | 2-15 | Introduction |
| HW 5 | 2-22 | Methods |
| HW 6 | 3-1 | Results and Discussion |
| Midterm 1 | 3-12 | Full Written Report |
| HW 7 | 3-22 | Abstract |
| HW 8 | 3-29 | Introduction |
| HW 9 | 4-5 | Results and Discussion |
| Midterm 2 | _4-16_ | Full Written Report |
| HW 10 | 4-26 | Abstract |
| Final | 4-30 | Full Written Report |
You can also gain extra credits if you attend scientific talks. This is described here.
This opportunity will allow you to earn up to 5 extra credit points on a Homework per week. These points can push you above 100% or help make up for missed exercises. In order to earn all points you must:
Approved talks: Talks given by researchers through the following clubs:
If you have any questions please consult Jeremy Rebenstock, rebensto@msu.edu.
All the material on extra credits is at https://github.com/mhjensen/Physics321/blob/master/doc/Homeworks/ExtraCredits/.