I think that we can all agree that Girl Scout Cookie season

I think that we can all agree that Girl Scout™ Cookie season amounts to a license for the Girl Scouts of America to print money. To meet demand and maximize profit, the dough is emptied from the mixer onto a high speed conveyor belt where it is cut into shape before being delivered to the oven. The conveyor carries the dough at 0.8c relative to you standing on the factory floor. While the dough is moving, a 5-cm square cookie cutter (at rest with respect to you) is pressed down such that the front and back blades simultaneously cut the dough and are immediately pulled back up. After the conveyor belt carries the cookies out of the oven and stops, it becomes clear to you that the cookies are not square. The width of the cookies (along the width of the conveyor belt) is still 5-cm, but the length of the cookies (along the direction of motion of the conveyor belt) is not what you expected. (a) You realize that the moving dough must have been length contracted relative to the cutter when it was cut. When you hold the finished product in your hand would this make the cookie too long or too short? What is the length of the cookie? (b) Alternately, you realize that from the dough’s reference frame, the distance between the front and back cutting blades must have been length contracted when the dough was cut. When you hold the finished product in your hand would this make the cookie too long or too short? (c) Likely, you have encountered a paradoxical result in the two preceding descriptions. Please provide the resolution. (d) If you really want to challenge your brain, explain what happens while the conveyor belt carries the dough through the oven.

Solution

The cutter sees the dough to be length contracted. Given the speed ofv= 0.8c, the relevant contraction factor is

= (1v2/c2)^1/2= 5/3

. Thus if the (moving) length of dough that is cut is L _moving= 5cm in the cutter frame, this corresponds to a length of

L_rest= L_moving= (5/3)(5cm) = 8.3cm. Thus the cookies are too long.

( 2)

Examining some of the finished cookies, you notice a problem: t hey’re not square! You measurethat the cookies are still 5 cm wide, as they should be, but the y are the wrong length (where width is the direction across the conveyor belt and length is the direction along it). Finally you think about how the cookies are seeing things and realize that as the cutter appears to be rushing over it (from the dough’s point of view), it is length contracted. What length of dough is actually getting cut? Are the cookies too long or too short?

From the dough’s point of view, it’s just sitting there while

the rest of the bakery rushes by around it, including

the cutter. Thus it says the cutter is moving, and thus is length contracted. Although the cutter thinks it is L_rest= 5cm when at rest, the dough says it is L_moving=L_rest/= (5cm)/(5/3) = 3cm long ( has thes ame value as above since the speed is the same). Thus only 3 cm are actually cut, and the cookies are too short.

(c) But wait! We’ve just said the cookies are both too long and too short! That’s not possible. At the end you

pick up a cookies and it’s either one or the other. It can’t be both. That means someone above is wrong. But

we used the same laws of physics for both people, and relativi ty is founded on the assumption that physics

must be the same for both.In the frame of the cutter, the dough is contracted and thus the cutter must be cutting it to be too long. In the frame of the dough, the cutter is contracted. The problem is that we assumed the cutter cut the

dough all at once. We assumed that the front and back ends of the cutter hit the dough at the same time. This, in

fact, is our error. Although the two ends do hit simultaneously in the cutter’s rest frame they don’t do so

for the dough. Whether or not two events are simultaneous depends on who observes them.In this case, the dough sees the front edge of the cutter hit first, then later the back edge, with the cutter

appearing to roll along the dough in between. Now the cutter is indeed only 3 cm long in the dough’s frame,

but during the delay between the front and back hitting, the dough has moved forward more. Thus more dough has gotten past the back of the cutter by the time it finally hits

. Showing correctly how much got through in the frame of the dough takes a bit more math than we’re getting

into in this class, but low and behold the answer is that the dough says 8.3 cm got through by the time it was cut. Thus the cookies are definitively too long.