Probabilistic Rolling

Classical Mechanics Level 4

If a point is chosen at random on a disc which is in a pure rolling motion on a horizontal surface, what is the probability that the velocity of that point is greater than the velocity of the center of mass of the disc ?

If your answer is $a$ then give you answer as $\left \lfloor 10a \right \rfloor$ .

The answer is 6.

2 solutions

Steven Chase
Mar 9, 2019

The diagram below shows the region of points that are moving faster than the center of mass. These points account for about $60.9 \, \%$ of the disk area. The edge of the disk is also plotted for display purposes.

@Steven Chase Sir, how did you come up with the exact equation of the region given??

Aaghaz Mahajan - 2 years, 3 months ago

Does Latex work for you today? It's not working for me. I was going to add more to the solution

Steven Chase - 2 years, 3 months ago

Yes, I think so...... $x^2 hello.... \sum_{n=0}^{99}\frac{x^n}{n!}$

Aaghaz Mahajan - 2 years, 3 months ago

It seems to work fine with me....

Aaghaz Mahajan - 2 years, 3 months ago

Here is the process for getting the equation
1) Write out the x and y velocities for a point on the disk (as functions of radius and angle)
2) Find the expression for the square of the speed, and set equal to the square of the translational speed
3) This gives the boundary curve in polar coordinates
4) Transformation to Cartesian coordinates indicates that the boundary curve is part of a circle

Steven Chase - 2 years, 3 months ago

When rolling, the contact point is the instantaneous axis of pure rotation. This fact makes it obvious that the boundary curve is just a circle centered on the contact point.

Nathanael Case - 2 years, 2 months ago

Laszlo Mihaly
Mar 11, 2019

The line where the disc touches the ground can be considered as an axis of rotation for the disc. The velocity of each point can be expressed as $v=\omega r$ , where omega is the angular velocity and r is the distance from the axis of rotation. For any given velocity the points that move with that velocity are on a circle (see the red and yellow lines). The yellow circle (segment) that includes the center of the disc separates the $v<v_0$ and $v>v_0$ regimes. The area above the yellow circle segment is 60.9% of the full area.

Probabilistic Rolling

The answer is 6.

2 solutions

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