Question

Special Relativity

1) (10 pts.) In a cookie factory, cookics are made in the following way: A circular cookio cutter comes down vertically and (very very quickly) stamps out cookics from a continuous layer of dough that moves past it on a convcyor belt. One day, due to a malfunction, the conveyor belt starts moving at relativistic spceds. Below are two statements, only one of which is correct. Identify the correct one, and explain why the other statement is incorrect: Statement 1: From the point of view of the conveyor belt, the cookie cutter is length- contracted along the direction of motion of the conveyor belt, and thcrefore it stamps out cookies that are compressed along that direction (the belt is eventually brought to rest, but that won't change the shape of the cookics in thcir own frame) Statement 2: From the point of view of the cookic cutter, the cookic dough has been length-contracted. The cutter still stamps out circular cookics from the length-contracted dough, so once the conveyor belt is brought to rest, the cookies will be stretched along the belt.

Answer 1

The statement 2 is the correct statement. The cookies are stretched out suppose, the cutter moves to the left. The right side of the cutter stamps the dough, then nearby parts of the cutter stamp it, and goes on, until finally the left side of the cutter stamps the dough. But by this time, the front of the cutter has moved further to the left so the cookie turns out to be longer than L (Let L be the diameter of the cookie cutter) Consider the moment when the right most point of the cutter strikes the dough. In the conveyor belt frame, a clock at the right side of the cutter reads L gamma /c^2 more than a clock at the left side. The left clock will advance \r\n by L gamma/c^2 by the time it strikes the dough. But due to time dilation, this takes a time r(L gamma/c^2) in the dough frame