# Student: Physics of rolling dice

Hello,

I'm a student currently nearing the end of high school, so my knowledge of physics (or even general math for that matter) are somewhat . . . lacking. That said, I'm currently attempting to figure out the various simple principles that apply to rolling a pair of dice across a table (for a self project).

What laws would apply to a set of dice as they are thrown from a hand, with a known rotation speed and a known velocity, onto a table? Could somebody please point me in the right direction as far as a collection of these rules is concerned, considering my skill level?

I've taken a high school course on simple physics, so I know the basics such as gravity and simple energy laws; hopefully that will help me understand the more advanced principles that apply to this scenario.

Thank you very much for any help! Edit: Ugh; sorry about not posting this in the "coursework" section. Although it's not really a homework question, I suppose I'd get more help there. Whoops Last edited:

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Danger
Gold Member
I'm no expert in this by any means, but I seriously don't believe that such a thing is calculable. There are just too many variables. Basically, each die has a 1/6 chance of landing on any particular number (assuming that they aren't 'loaded').

AlephZero
Homework Helper
I'm no expert in this by any means, but I seriously don't believe that such a thing is calculable. There are just too many variables. Basically, each die has a 1/6 chance of landing on any particular number (assuming that they aren't 'loaded').
It's certainly calculable, assuming you have good enough assumptions about the material properties and friction behaviour. There are several commercial available computer programs for dynamics simulation that could do it easily enough (e.g. DYNA3D and its various clones and imitations)

But all you would demonstrate is that the result is very sensitive to the initial conditions - so unless you want to do something like investigate if there are any throwing techniques than could be used to "cheat" by biasing the probabilities, or something similar, there's not much point.

There are just too many variables.
This is eventually going to be for a computer application, so any variable which is required can be known. In fact, if something is just too difficult to calculate (such as air currents), it's not hard to say that this takes place without air. All I'm trying to find are the equations which I must use to account for all of the different movements of each die.

But all you would demonstrate is that the result is very sensitive to the initial conditions - so unless you want to do something like investigate if there are any throwing techniques than could be used to "cheat" by biasing the probabilities, or something similar, there's not much point.
Even if there's no point, I'd still like to investigate it I've been looking through my course notes for last semester's physics course and have found plenty of laws which would apply to this scenario, but I can't help but think that most of the things which we were taught only apply to very basic scenarios, unlike this one.

Are there any specific equations that I should look for?

AlephZero
Homework Helper
The individual events (each separate impact with the table) are pretty basic. The messy part is stringing a series of the events together - which is what computers are good at.

I would suggest starting with a 2-D version of the problem. Consider a square object falling onto a plane with an arbitary linear velocity, angular velocity, and orientation when it hits the table. Find the conditions for it to slide on one face, bounce back into the air, start to "roll" along the table. You should be able to make a model of that using conservation of linear and angular momentum, the coefficient of restitution at the impact, and the friction coefficient (assuming Coulomb's friction law).

Another simple case would be to assume inelastic collisions and a high enough coefficient of friction, then look at the dice "rolling" along the table - e.g. given the intial horizontal velocity, how many times does it rotate before it loses too much energy and stops. That's a version of the standard textbook question about "when you push a box sideways, does it slide or tip over".

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In our course we only briefly touched on 2-D physics (most of it was 1-D), but here's what I understand so far:

I would suggest starting with a 2-D version of the problem. Consider a square object falling onto a plane with an arbitary linear velocity, angular velocity, and orientation when it hits the table.
Alright, I'm guessing that when you talk about "linear velocity", "angular velocity", and "orientation when it hits the table", you're talking about these three things, respectively: Find the conditions for it to slide on one face, bounce back into the air, start to "roll" along the table. You should be able to make a model of that using conservation of linear and angular momentum, the coefficient of restitution at the impact, and the friction coefficient (assuming Coulomb's friction law).
Not quite sure what any of these are. Does "conservation of linear and angular momentum" mean that it continues to rotate as quickly as it did before it hit the table, and that it bounces back up with the same energy that it hit the table with? I know that the coefficient of friction is the constant applied to the friction equations, but I have no idea what Coulomb's friction law is.

That's a version of the standard textbook question about "when you push a box sideways, does it slide or tip over".
I understand exactly what you mean, but I've never seen that question before and wouldn't be able to tell you how to begin Coulombs friction just relates the Normal force and retarding acceleration by a constant, mu, probably same as what you have seen before.

The coefficient of restitution is a measurement of the elasticity of a collision. A superball has a high C.O.R., a die has a low one- this is the ratio of velocities before and after impact. Conservation of linear and angular momentum is what you suggest, but subject to COR correction during impact and soon after it, friction. Its a bear of a problem to be taking on with so little experience; I admire your gumption.

Its a bear of a problem to be taking on with so little experience; I admire your gumption.
The more I can learn, the better I'm slowly beginning to put together the various factors involved. The description you gave was very clear, thank you. Here's my understanding of what you've described of C.O.R so far: ...comparing a die versus a super ball.

So, piecing together this information, I'm guessing that the equation must be something like:

$\vec{v_2} = c * \vec{v_1}$

Where v1 is the initial velocity, c is the C.O.R, and v2 is the final velocity, where -1 <= c <= 0. That's just my guess at how to model that, I'm sure the actual equation is much different :tongue2:

I'd also need to use:

$v_i = gt$

I'm not quite sure how to start the calculations for angular velocity, though. I've looked at wikipedia, and it gives this equation:

$\omega=\frac{|\mathrm{\mathbf{v}}|\sin(\theta)}{|\mathrm{\mathbf{r}}|}$

...but I really don't understand what that means.

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The angular velocity is actually a vector, which is defined by the following relationship:

$$\ \vec[v]=\vec[\omega]\times\vec[r]$$, where each component is the velocity of rotation around an axis perpendicular to the x, y, and z axis (arbitrary axis can be chosen to make the mathematics more manageable)

In the 2D case you are investigating, imagine an axis perpendicular to the plane in which the object is rotating (This is the direction of this component of the angular velocity), with magnitude omega. Then $$\vec[v]=\vec[\omega]\times\vec[r]\rightarrow \left|v\right|=\left|\omega\right|\left|r\right|sin(\theta)$$, where theta is the angle between the two vectors (Omega and r, which in this case is 90 degrees). The bars denote the Euclidean norm, or the "length" of these vectors.

omega is the rate of rotation in rad/s. r is a vector extending from the origin of the point of rotation to the point who's angular velocity you are interrogating, and v is the tangential velocity vector of that point at that given instant. Here since r and v are coplanar, omega points out of the page (call this the z direction), and you can solve for omega, and get that formula from wikipedia.

In general, rotational dynamics are pretty complicate as what is called the body's "Moment of Inertia Tensor" and angular velocity vector both change with time (you can visualize this), thank of the inertia tensor as a matrix which tells you the tendency or willingness of the object to rotate around a given axis.