Glue a quarter (the coin) to the top of the desk?

Question

Place another quarter flat on the surface of the desk, press its edge against the edge of the first quarter, and start roll it with a finger around the first quarter without skidding of their edges.

If the center of second quarter completes one revolution in 3 seconds, what is angualar velocity of this qurter with respect to the desk?

Answer 1

"The moving quarter rolls, and having rolled, moves on..."

[An amusing question!]

The answer is 4π/3 radians/s = 4.1888... radians/s, or alternatively 240 deg/s.

By the time the moving quarter is on the OPPOSITE side of the fixed quarter, it will have completed one rotation, HALF of its circumference having rolled on HALF of the (curved) circumference of the stuck quarter. (In effect, because the surface being rolled on has changed ITS own direction by π radians, the rotation of the rolling quarter relative to that curve only has to add a further π radians to it, to complete one full spatial rotation.)

So its angular velocity is 2π radians in 1.5 seconds

= 4π/3 radians/s = 4.1888... radians/s, or alternatively 240 deg/s.

Live long and prosper.

Answer 2

Because both quarters have the same radius, the angular velocity of the edge of the second quarter to its center will be the same as the angular velocity of the center of the second quarter to the center of the first quarter.

Think about it. In one revolution, the quarter will roll out one circumference of distance. Since it is confined to the other quarter's circumference, it must roll out this distance along the circumference of the glued quarter. But the circumference of the glued quarter is the same as that of the rolling quarter. The effect is that each time you roll the quarter one revolution, it will be at the same spot as when you started. If you don't believe me, get out some quarters and try it, but the math should be sufficient to agree with the statement.

Therefore, it takes an equal amount of time for the center-edge rolling quarter system to trace out 2*pi as it does for the center-center rolling quarter and desk system to trace out 2*pi. Since you gave that the center-edge rolling quarter system (I assume that's what you meant. But if you didn't, luckily it doesn't matter) has a frequency of 1/3rev/s, we can find angular velocity by the equation w=2*pi*f

So

w=2/3*pi rad/s.

EDIT:

Dr Spock is incorrect when he says it traces out twice its own circumference in one quarter-quarter revolution. It is half of his answer. Dr Spock, I encourage you to pull out 2 identical coins for yourself. First, try holding one down and going around as is said. This will prove it definitively. Then, however, you should put them side by side (say with the eagle head pointing away from you), and rotate them both oppositely at the same time. You'll then begin to see WHY they both end up the same at the same time. Alexander said to make the quarters not slip. When you have them side by side, you see that the only way not to have one slip is to make one go faster than the other. But you can't decide which one is to go faster, since they're identical in this frame. Then there can be no frame in which one is rotating faster than the other while simultaneously not slipping. Further, even if there were a possible frame, we've shown that it's impossible to determine which one would rotate faster, since in the side by side frame, they have to rotate at the same rate, like gears of equal size. If you can't tell which should go faster, it would be impossible for either to be determined to go faster relative to each other.

Answer 3

The center of the coin has velocity v. This center is revolving about hte center of the fixed coin ie with a radius 2r, so
2π/3 = v/(2r)
v/r = 4π/3

Since the point of contact between the coins has zero instantaneous velocity,
ω = v/r = 4π/3

Answer 4

A circle has 360 degrees, right?

To find the rotation rate, or angular velocity, you will need to divide the number of degrees by the number of seconds

Angular Velocity =
Total number of degrees traveled/Total time it took to travel

Dr. Spock, it is "the center of the second quarter completes one revolution", not "the second quarter completes one rotation".

Answer 5

Before attempting mathematics, learn to spell and use grammar correctly.

Answer 6

$1.50?