Phisics-Engine

This repository is going to implement a simple physics engine for 2D point mechanics. I am going to use the Euler-Richardson algorithm (RK2) for simplicity. The simple animation is going to go on the browser canvas. The project is going to be part of a mathematics-physics training project for elementary and secondary school pupils.

Content

1. Quick Start Guide
2. Sign Up
3. Log In
4. Create Project
5. Execute Project
6. Numeric Results
7. Time Graphs

Quick Start Guide

I recommend first to create an account with your email address form the Sign Up menu. Without creating an account, you still can try the projects and even modify them, but the modifications are local to your computer. You also don’t get refreshed when someone modifies their projects, unlike with an account, because the projects get updated whenever you Log In.

After registration, you need to verify your email and you ready to execute the already existing projects or create your own ones. You are also able to publish your projects and share them with other members.

To execute a project you need to go to the home page, where you can navigate between projects with the Next and Back buttons. Chose your project by clicking on its link then when the page loaded press the Start button. You will see an animation, which is the physics simulation. You may see the elapsed time and energy values at top-right corner of the animation window and you can stop it and reset, by the appropriate buttons.

The simulations use metric units (m, kg, s, N, J, etc.) for normal simulations and astronomical units for the celestial mechanics simulations, which show planetary orbits around the Sun. For these simulations the unit of time is year and the unit of distance is the astronomical unit, which is the Sun-Earth mean distance (about 150 000 000km). The mass unit is the mass of the Sun. In these units the gravitational constant in Newton’s gravity law is about $4\pi^2$ .

You can also see a bunch of check-boxes, which control what it shows in the animation window. You can turn off/on the time and energy values in the top-right corner, the unit grid, the force vectors on the bodies and the trajectories of the bodies.

You can get numerical results by ticking the Results checkbox and you are also able to get time graphs by ticking the Graphs checkbox. The app is able to show 4 different graphs on one page. First you need to pick the maximum 4 different quantities from the numerical results by choosing them with the selector and ticking the Added to graphs check-box. You can remove them from the graph from the similar way with unticking the checkbox.

After the selection, you need to tick the Graphs check-box and run the simulation by clicking the Start button. The graph slowly appears as the time elapses.

To create a project you need to click the Create menu in the top navigation and it opens the page. Here you can edit the current project or press the Create button to create a new one. This leads to an empty world with some predefined values, which are appropriate for most normal simulation projects.

You can create new points from the points part with the New button and delete them with the Delete button. All the points have an associated number (index) and changing that, we can navigate between the mass points (bodies). We can also edit the parameters of the point, with the given index. The Point fixed checkbox creates fixed points, which cannot move at all.

The coordinates are in length units and the top-left corner of the simulation window is always the origin. When the x position is growing the point moves to the right, when y is growing it is moving down. The gravity of Earth is pointing down to the y direction.

Caution: The default value for g is 0, so there is no gravity. You must give the 9.8 parameter for g for normal simulations.

The rod is a stiff spring. You can create them by the New button delete them by the Delete button, similarly to points. You can adjust the stiffness and dumping of these. You also need to specify the indexes of the end points, which the rod connects. All rods have their own index and by ticking the Is spring checkbox the graphics will remind for a spring during the simulation.

From these elements you can build complex simulations. You can also create periodic force, which effects some point to simulate resonance phenomenons. The frequency of this force can be adjusted between the minimum and maximum value by the range input on the project page meanwhile you are executing the simulation.

At the end give a name to the project and a short text description mainly for accessible users, who may not be able to see the animations. Do not forget to press the Save button to save the new project.

You can publish your project for the other registered users with ticking the Published checkbox.

Caution: You may get error messages if you are not logged in. The browser still saves into local storage, but this gets overwritten if you log in to your account.

Sample Project I

The period of the mathematical pendulum is independent of its amplitude for small amplitudes. What is happening to the period if the amplitude grows and not small at all?

To answer this question and practice a bit all of the above, I will make a sample project. We will make two mathematical pendulums, which can hang independently and we can show and compare their x position, depending the time. This way we can answer the question.

1. Let’s log in to our account.
2. Create a new project (Navigate to create page, press Create button)
3. Do not forget to set g to $9.8m/s^2$
4. Create a fixed point to hang the pendulum(s) at x=3m y=0 position (Press New at points, tick Fixed point checkbox, fill in x and y)
5. Create a mass point at the x=3m y=3m position and let’s give it vx=0.1m/s vy=0 initial velocity. The m should be 0.1kg (Similar like previously without ticking the checkbox)
6. Connect the two points (index=0 and 1) with a new rod (Press New button at rods, make sure elasticity = 10000, beta=100, point1 is 0 and point2 is 1 and length=3m, the Is spring checkbox is unticked)
7. Fill in the title and text description and press Save button
8. Run it (Navigate to project page, press Start) It should be hanging nicely with tiny amplitude
9. Press Stop button and navigate back to the create page
10. Create another mass point at x=3m, y=3m position with the same m=0.1kg mass. This has index 2. The vx=5m/s vy=0 are the initial velocity.
11. Add another rod, which connects the points with 0 and 2 indexes. So point1=0, point2=2, elasticity=10000, beta=100, length=3m.
12. Run it again the same way like in 8. The two pendulum is not hanging perfectly synchronously.
13. Press Stop button and tick Results checkbox
14. New surface appears, navigate to point index 1, choose x position and tick Added to graphs checkbox
15. Do the same like in 14. but this time with point index 2
16. Press switch back button to make this surface disappear
17. Press Reset button to reset the simulation
18. Tick the Graphs checkbox
19. Press Start button, the graph should appear slowly. It’s clear from the graph that the pendulum with the large amplitude has slightly bigger period. This answers the question, so the period of the mathematical pendulum grows with the amplitude.

Sample Project II

There is a 0.1kg mass point, which can slide on a horizontal surface. It starts in a rest position. There is a 1N constant horizontal force accelerates it for 1s then the friction slows it down until it stops. How much is the achieved speed at the acceleration? How long is the path until it achieves this speed? How long it takes the friction to stop the mass point again? How far the mass point gets from the starting point? The friction coefficient is 0.3.

Let’s solve the problem with a calculation first. The normal force balances the gravity:

$$N = mg = 0.1\times 9.81 = 0.981N$$

The friction force:

$$F_f = \mu \times N = 0.3 \times 0.981 = 0.2943N$$

The accelerating force:

$$F = F_c - F_f = 1 - 0.2943 = 0.7057N$$

The acceleration: $$a = F/m = 0.7057/0.1 = 7.057m/s^2$$

The achieved speed:

$$v = \Delta v = a\times t = 7.057 \times 1=7.057m/s$$

The length of path at acceleration:

$$s_1=\frac{a}{2}t^2 =7.057/2 \times 1=3.5285m$$

The deceleration from friction:

$$a=\frac{F_f}{m}=\frac{-0.2943}{0.1}=-2.943m/s^2$$

The elapsed time until stop:

$$t=\frac{\Delta v}{a}=\frac{-7.057}{-2.943}=2.3979s$$

The length of path during deceleration:

$$s_2=v_0t+\frac{a}{2}t^2$$ $$s_2=7.057\times 2.3979+\frac{-2.943}{2}\times 2.3979^2$$ $$s_2=8.4610m$$

The full path:

$$s = s_1 + s_2$$ $$s = 3.5285 + 8.461 = 11.9895m$$

Let’s make the project:

1. Create project (go to create page and press Create button)
2. Add title and short description and save it (press Save button)
3. Add 9.81 to g
4. Change scale to 0.025, dt to 0.000001 and anim time to 0.003333333333333333
5. Make a fixed point at x=0, y=3 (Click New button at Points and input the coordinates then tick Point fixed checkbox)
6. Make a fixed point at x=15, y=3 (Do the same as in 5, but x=15)
7. Make a rod between points 0 and 1 (Click New and rods, make sure that length is 15 and point1 is 0 and point2 is 1. Also make sure that elastic constant is 10000 and beta 100)
8. Make a sliding point at x=0, y=2.9. m=0.1 and it is not fixed. (Press New at Points then give all the details and make sure the Point fixed checkbox is unticked)
9. Turn on point-rod collisions and add friction (tick the body-rod collisions on checkbox and add 100 value point-rod coll. beta. You also add 0.3 to point-rod coll. mu)
10. Create a 1N constant external force, which effects the 2nd point for 1s (Set the following data in periodic external force: point = 2, t min = 0, t max = 1, freq min = 0, freq max = 0, F0x = 1, fix = 1.570796, F0y = 0, fiy = 0 then tick the Is on checkbox)
11. Tick the show time, show grid, show force and show energy checkboxes to visualize data in simulation window
12. Run the simulation (navigate to project page and press Start button)
13. You can see that the point gets to rest after about 12m horizontal movement, which takes about 3.5s. Check this exactly. (press Stop when the sliding point is in rest, choose point 2, you can see x or path length is 11.9805 in exact agreement with our calculations)
14. Make t graphs of x, vx, ax od point 2 and Fx from periodic external force. (Press Stop then Reset buttons. Tick Results checkbox. Choose point 2. Pick x and tick Added to graph checkbox. Do the same with vx and ax. Choose periodic external force. Choose Fx and tick Added to graph checkbox. Tick the Is origin centered checkbox. Press switch back button. Tick Graphs checkbox and press Start button) You can see the above graphs slowly appear.

Sample Project III*

Caution* This project is for people, who know well the basic trigonometry.

There is an incline, which has 30degrees with the horizontal direction. How far the mass point goes on the incline when its initial speed is 3m/s? How high it gets when it turns back? How much is the magnitude of its acceleration and the sum of the forces effect it? How much is the normal force? The friction is negligible and the mass of the point is 0.1kg.

Let’s first do a calculation and we can verify it by a simulation. The sum of forces is:

$$F=mgsin\alpha$$

In this case it’s:

$$F = 0.1 \times 9.81 \times sin 30=0.4905N$$

The magnitude of acceleration:

$$F = ma$$

$$a=F/m=0.4905/0.1=4.905m/s^2$$

The time of deceleration is the same as the time of acceleration,so

$$a = |\Delta v|/t$$

$$t=|\Delta v|/a=v/a=\frac{3}{4.905}=0.61162s$$

The length of path during this time

$$s=a/2\times t^2 = 4.905^2/2\times 0.61162^2$$

$$s=0.9174m$$

The achieved height is

$$h = s\times sin\alpha = 0.9174 \times 0.5$$

$$h = 0.4587m$$

We should get these details from a simulation too. Let’s make the simulation. The first endpoint of the incline will be x0=1m and y0=3m. Let’s choose the length of the incline l=2m. Let’s calculate the other endpoint:

$$x_1 = x_0 + l\times cos\alpha$$

$$x_1 = 1 + 2cos 30 = 2.732051m$$

$$y_1 = y_0 -l\times sin\alpha$$

$$y_1 = 3 - 2sin 30 = 2m$$

Now we calculate the position of the mass point at the start, when its radius = 0.1m.

$$x_2 = x_0 - r \times sin\alpha$$

$$x_2 = 1-0.1msin30 = 0.95m$$

$$y_2 = y_0 - r \times cos\alpha$$

$$y_2 = 3-0.1cos30 = 2.9134m$$

We also need to calculate the component of the initial velocity:

$$v_x = v_0 \times cos \alpha = 3cos 30 = 2.598076m/s$$

$$v_y = v_0\times sin\alpha = 3sin30 = 1.5m/s$$

Now we have all the details for making the simulation.

1. Create a new project (Choose Create menu at top bar then press Create button)
2. Add 9.81 to g
3. Press the New button at points to create the bottom point of the incline. Set x=1 and y=3 then tick Point fixed checkbox.
4. Press the New button again to create the top point of the incline. Set x=2.732051 and y=2 then tick Point fixed checkbox again.
5. Now press the New button at rods to create the actual incline. Make sure it has point1=0 and point2=1, so it connects the previous points. The length=2 and elasticity=10000 and beta=100.
6. Press New button again at points to create the sliding mass point. Set m=0.1 x=0.95 y=2.9134 and tick Is path visible checkbox to show the path. Set vx=2.598076 and vy=-1.5. vy must be negative, because the point goes up but y grows downwards.
7. Name the project and write a short text description.
8. Tick the show time, show grid, show force and show energy checkboxes to see more data during the simulation.
9. Do not forget to tick the body-rod collision on checkbox. This makes sure, that the point won’t ignore the incline, so the system calculates the forces between them. Make sure that collision k parameter is 10000 and point-rod coll. beta is 100. These are the elasticity and damping constants of the collision between the rod and the mass point.
10. Click the project link in the top bar and the Start button to run the simulation. It runs quite fast.

Tip The project runs quite fast, but we can slow it down by changing the dt and anim time parameters in the same ratio. If for example we choose dt=1e-7 and anim time=0.0003333333333 it slows down everything 100 times. We should keep the ratio of these parameters the same as the original setup was.

11. Now we can get some graphs. Press Stop button and Reset button then tick the Results checkbox.
12. Choose index 2 for the point then select path length and tick Added to graphs checkbox.
13. Select v then tick Added to graphs checkbox again
14. Click switch back button and tick Graphs checkbox then click the Start button to get the graphs. We get the below graph:

The red curve is the path length, which changes very slowly around the time, when the body turns its direction of movement. The green v-shaped line is the speed. It goes down to zero and up when the body goes back to its original position. The turning point is when v=0. We see that the values, what we can read from the graph are in consistency with our calculation, although graph readings have limited accuracies.

15. Untick the Graphs checkbox then press Reset and Start
16. Press Stop when the time is 0.611 roughly
17. Tick Results checkbox and choose 2 for point index. The sliding point is red now, which means it’s selected and we can see its numerical values. Thepath length=0.9174, a=4.905 and v is nearly 0 in full agreement with our calculations. We can read y=2.4548.

$$h=|\Delta y| = |y - y_0|$$

$$h=|2.4548 - 2.9134|=0.4586m$$

This is again in full agreement with our calculations.

Sign Up

You can create an account from here, if you do not have one. You just need to fill in a form. You need a valid email address, because you will receive a verification email and you need to click on the link to verify your email address. The form to fill in has the following fields: email, password, confirm psw, first name, last name. The first and last names are optional.

Log In

Caution When you are logged in, but getting access denied errors, the validity of your token has expired. In a case like this you must log out at the home page and log in again.

The form has the email and the password fields. You also can get to the sign up page from here if you don’t have an account yet. At successful login you get a confirmation with a button, which redirects you to the home page. There is a link, for the case you have forgotten your password. It will send you an email with a link, which makes you possible to make a new password.

Create Project

Creating simulation projects means that we specify the general simulation parameters and the initial conditions of all the points and rods, which build up the simulations. Rods are very stiff springs. What kind of forces do we have?

1. Gravity It’s the force, which pulls downwards each bodies close to the surface of the Earth. This is the force we calculate with the formula $F_g = m \times g$ and this force always points downwards, which in our simulations the y direction. We can adjust the value of g as a general parameter. For celestial mechanics simulations there is an attracting force between each pair of bodies, which we calculate with Newton’s formula: $F_g = G \frac{m_1m_2}{r^2}$ In this formula $m_1$ and $m_2$ are the masses and r is the distance between them. The units are astronomical units, so $G=4\pi^2$, the distance unit is the Sun-Earth medium distance (150 000 000km) and the mass unit is the mass of the Sun. The time unit is the year.

Caution For celestial mechanics simulations you have to choose celestial mechanics for simulation type. The default value is normal simulation, which means experiments either in no gravity or close to the surface of the Earth as we have all got used to it.

Caution For celestial mechanics simulations you may need to specify the appropriate force multiplier. The number of the force in astronomical units can be tiny, like 5e-5, so we need to multiply it to make it visible when you try to visualize it in the simulation or graph.

2. Elastic and dumping forces between points, which are connected by a rod. The rod follows Hook’s law. $F_e = -k \Delta l = -k \times (l - l_0)$ The force is proportional with the deformation, the k constant is the elastic constant. The minus sign expresses that the elastic force tries to decrease the deformation. The damping force is a velocity dependent force, which depends on the relative velocity of the end points. If this relative velocity has parallel component by the rod, the damping force is $F_d = -\beta v_r$, where $\beta$ is the damping constant, $v_r$ is the relative velocity component, parallel with the rod. The minus sign expresses that its direction is so that it always decreases the relative velocity component.
3. Collision forces You can allow collisions between points and between points and rods. In both cases we imitate the collision force with elastic force. This is how we deal with the normal force on inclines for example. There is dumping on the calculation of these forces too.
4. Friction forces If you specify more than 0 for the $\mu$ (mu) either for point-point collisions or point-body collisions, we have the Coulomb-model of friction forces. The friction force is trying to decrease the relative velocity, so it points to the opposite direction and parallel with the surface (tangential direction). If the relative velocity is zero, we get static friction. The static friction can be calculated from the condition that the relative acceleration is zero, so the net force should produce the acceleration, which the surface point, If the friction is dynamic, the value of the force is $F_f = \mu \times N$ In this formula N is the normal-force and $\mu$ is the friction coefficient, which is usually maximum 1. The value depends on the surfaces mainly, both the materials and the smoothness of the surface effects it.
5. Periodic External Force We can also add some periodic external force to effect one point. This is mainly useful to check resonance phenomenons. This force starts at some point in time and stops at a later point in time. We define it by its components, like the following formulas: /

$$F_x = F_0,_x sin (\omega t + \phi_x)$$

$$F_y = F_0,_y sin(\omega t + \phi_y)$$

$$\omega = 2\pi f$$

Here f is the frequency, which is between the min and max values and it can be adjusted real time with a range input, when you run the simulation.

Tip You can add a constant force to a point for a certain amount of time, because f can be zero and the phase factors can make the sin functions 1.

6.Air resistance You can add air resistance to the points with specifying the drag coefficient above 0. It’s usually 0.47 for spherical bodies under normal circumstances. The force, effecting the spherical bodies, is /

$$F_d = \frac {1} {2} \rho v^2 C_d A$$

$C_d$ is the drag coefficient, usually 0.47 for smooth spheres, although it changes for very high speeds or very low speeds. It also depends on the shape of the bodies, hence the car can be designed to have a low drag coefficient to reduce air resistance. A is the area what the flow “sees”, so for a sphere it’s $A = R^2 \pi$ if the radius is R.

The forms, can be used to define the simulation parameters, the mass points or the rods are pretty straightforward. You need to specify all the points and rods for the simulations. The forms also specify the above parameters and the initial conditions. For celestial mechanics rods are not necessary, because the only important force between the points is gravity.

We still have to mention the Copy and Delete buttons, beside the Save and Create buttons. With the Copy button, we can copy an existing project as our own to modify it. The Delete button can delete our projects.

Execute Project

You navigate to the project page and press the Start button. You can also stop or reset the project with the relevant buttons. The Zoom In and Zoom Out buttons can zoom in and out of the world view, although the origin of the coordinate system stays in the top-left corner. We can turn on and off some visualisations with the checkboxes, like Grid, Time, Force and Energy. The Graph checkbox changes the view to show the graphs instead of the simulation. The Results checkbox is used to show a piece of UI, which shows all the important numerical results and you can also pick the quantities you can see on the graphs.

Numeric Results

You normally stop the project by clicking the Stop button and tick the Results checkbox. This opens up a piece of UI, where you can pick an element of the simulation (a point or a rod) or you just get an overview of the simulation parameters, like time or g. Whatever element you select, it’s going to be red in the simulation view. You can also view the collision and friction forces regarding the selected element. On the image, we can see the Sample Project II numeric results. The rod (incline) and its collision with the 2nd point (the sliding one) has been chosen. We can read the N (normal force) and the F (friction force).

Time Graphs

When you go to the Numeric Results you can choose maximum 4 parameters, which can be shown as the function of time on a graph. These can be anything available on the select element, like the x coordinate of the 5th point or the acceleration of the 6th point. It also can be the periodic external force or the normal force between the 0th rod and the 2nd point. When you have chosen the parameters, you can view the graphs by pressing the switch back button and tick the Graphs checkbox. This makes the animation disappear in the simulation window. If you press the Start button, you see the graphs gradually appear as the simulation progresses. The origin is the bottom-left corner, we can magnify by the Zoom In and Zoom Out buttons, but we cannot change the position of the origin, unless we tick on the results page the Is origin centered checkbox. This centres the origin on the y axis, so we can see if a parameter negative.

Caution The path is not going to be recorded when the simulation is drawing the graphs. If you untick the Graphs checkbox you will see inaccurate paths. You must reset and rerun the simulation to see the correct paths.