- #1

etotheipi

There’s a bus driving along a road at say, ##v \, \text{ms}^{-1}##, and on one of the tyres someone has painted a bright yellow dot. The tyres have radius of ##1/(2\pi) \, \text{m}##, or in other words, they complete ##1## revolution in ##(1/v) \, \text{s}##. Kip, who’s attached to the bus (don’t ask me how…), uses a stopwatch to measure the time taken for each revolution of the wheel. That is, he’s just pressing the lap button every time the yellow dot reaches the top of its cycle. Charles is instead standing still on the pavement, but is also measuring the time taken for each revolution in the same way.

Because the speed of the bus relative to the pavement (and similarly the speed of the pavement relative to the bus) is just ##v \, \text{ms}^{-1}##, Kip and Charles should both calculate that the wheel - which is rolling - has the same angular speed ##\omega = v/r =2 \pi v \, \text{s}^{-1}##.

But the thing is, the events “yellow dot is at the top of it’s ##\text{i}^{\text{th}}## and ##\text{j}^{\text{th}}## cycle” respectively are of course at the same position coordinates as measured by Kip, so it’s easiest to just apply the so-called ‘time dilation formula’ which will tell you that the time interval between ##E_i## and ##E_j## calculated by Charles should be larger than that calculated by Kip.

That seems weird to me, because the effect of time dilation appears to be “invisible” here. What’s the missing piece of the puzzle?