Dot Dot Dot Dot Dot Dot Dot Dot

Classical Mechanics Level 2

In the above gif, the white dots are each moving in a straight line along the spokes of a wheel of radius 1 meter. This gives the appearance of a white circle that moves around.

If the white dots complete 1 revolution in 1 second (back to the same starting position and velocity), what is the speed at which the center of the white circles moves?

\pi \text{ m/s}

2\pi \text{ m/s}

2 \text{ m/s}

1 \text{ m/s}

1 solution

Brian Charlesworth
May 1, 2015

For this situation to occur, the white circle's diameter must be the same as the black circle's radius, which is given as $1$ m.. The center of the white circle thus traces out a circular path of radius $r = \frac{1}{2}$ m. over one second, i.e., it covers a distance of $2\pi r = 2\pi*\frac{1}{2} = \pi$ m. in one second, and thus has a speed of $\large\boxed{\pi}$ m/s.

This has got me thinking.. What is the angular velocity of the white circle about its center? At first look it seems like the white circle is pure rolling inside the larger one..

Raghav Vaidyanathan - 6 years, 1 month ago

I'm thinking that it would be $2\pi$ rad/s, since from the perspective of the center of the white circle each dot makes one full revolution about it in one second. We can see the speeds of the dots vary as they move back and forth on their respective lines, but I think that from the perspective of the center of the white circle they appear to be moving uniformly. Does that sound right?

Brian Charlesworth - 6 years, 1 month ago

Are the white dots doing SHM ?

Kushal Patankar - 6 years, 1 month ago

Assuming that the center of the white circle is moving uniformly, then the white dots has SHM motion. Do you know how to show this?

Hint: SHM is achieved when we project a point that is moving at constant speed on the circumference of a circle down onto an axis.
Find such a point, and relate it to the movement of a specific white dot.

Calvin Lin Staff - 6 years, 1 month ago

@Calvin Lin This gif is hypnotizing. :) Just for sake of absolute clarity it might help to specify that it is a "radius $1$ meter wheel", (although if we took the diameter as $1$ m. then we wouldn't get any of the answer options).

Brian Charlesworth - 6 years, 1 month ago

Thanks. Updated the phrasing.

Calvin Lin Staff - 6 years, 1 month ago

Well done sir! Upvoted!

Sravanth C. - 6 years, 1 month ago

Dot Dot Dot Dot Dot Dot Dot Dot

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