Tag Archives: Clocks

A short post on length contraction

20 Mar

Here’s a quick recap of the things we know about relativity; the perception of time is dependent upon velocity, and the speed of light is a constant regardless of the frame.(check out the full episode #1 here and #2 here). Alright, so we know something about the time and we know something about the speed. If you’ve ever dealt with rates before, you might be wondering “What’s happening to the distance?”

That’s a fair question, but it’s a little more complicated than it appears on the surface. Let’s think back to our train problem. If we want to measure the train at rest, we can hold a tape measure up to the ship and record the length. That’s works because the ship and the tape are both at rest with respect to one another, but what if we wanted to measure the space ship while it’s moving?

Now we have a problem. We could hold the tape measure still and wait for the train, but we’d have to be able to make a length measurement instantaneously as the train reached the tape. It’s not a good plan. We could move the tape measure at the same speed as the train, but then we’re taking another measurement with the tape at rest with respect to the train. Hmm. Let’s try something different. Bring in the timer gates.

image095

Here’s a quick refresher on timer gates. They are a gate with a laser between them (here represented by the dashed lines). When an object passes through the gate and breaks the laser, an electrical impulse is sent down the black wire to the computer which records the time. That’s all there is to it.

How can we use these? If we set up two along the tracks of the moving train, we can use one to record when the back of the train goes by and the other to record the arrival of the front of the train. You could do some math to determine the length of the moving train as a function of the time difference and the distance between the gates…but you probably don’t like math. The other option is moving the gates until the back of the train goes by one gate at the exact time that the front of the train arrives at the other.  At that point, the distance between the gates is the length of the moving train.

Alright, so I do that experiment and I find that the length of the moving train is related to the resting length of the train by this equation

image009This means that the moving object appears to be shorter than the object at rest by some factor related to the speed at which it’s moving. Actually, that factor seems kind of familiar. Didn’t we see that when we were dealing with time contraction?image008Yep. Same factor. That’s because it needs to be in order to balance out the equations that makes the speed of light a constant. Cool.

Two last notes about length contraction.

#1. The length contraction would be perceived by both the people in the moving object and the people not in the moving object. Each would consider themselves at rest and the other objects moving.

Length Contraction

On the top: Trains DEF seem short from the frame of ABC
On the bottom: Trains ABC seem short from the frame of DEF

#2. The length contraction only occurs in the direction of motion. So, in our last example, the height of the train would remain the same even while the length was apparently decreasing.

That’s a pretty good primer on length contraction. I’ll do one more post on relative mass and then we’ll take a little break from physics to look at some other cool stuff.

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Keep moving, you’ll stay younger

16 Mar

So in the last episode, I explained very basically what relativity means. We saw that the same event could appear differently to two observers depending each man’s frame of reference. That was bizarre, and from there we could begin to ask other questions. For instance, what other measurements depend on our frame of reference?

Let’s start with one of the more basic measurements-time. We’re going to use another example to explain what’s happening, but this time we’re also going to use some algebra and a triangle. Bear with me, the math isn’t too bad.

Source: PhysicsForIdiots.com

Source: PhysicsForIdiots.com

Alright, so imagine that we’ve got a space ship that’s traveling at speed v, and there’s a flash of light that is bouncing up and down inside of the space ship. The speed of  the flash of light (and all light for that matter) is c, and that doesn’t change because the speed of light is a constant. So far so good? Great.

Source: PhysicsForIdiots.com

Source: PhysicsForIdiots.com

For an observer on the ship, we’re at rest with respect to the ship. Another way of saying that is that the ship is his rest frame. If we call the height of the ship L, then the total time it takes the light to travel from the floor to the ceiling is

image004

Source: PhysicsForIdiots.com

Source: PhysicsForIdiots.com

But looking up at the ship from Earth is a different story because we’re dealing with a different frame of reference. While the light is bouncing up and down, it also has to travel horizontally with the rest of the ship. The total distance it travels is the diagonal, D. As a result, the time it takes in this alternate frame is.

image005

Note that the speed of light, c, is the same for both of these equations. D, however, is not the same as L; D is longer than L.

Let’s stop right here to drive home that concept. The light travels less distance in the rest frame than it does in the moving frame. As a result, less time passes in the rest frame than in the moving frame. A clock on the space ship would go slower than a clock on Earth. That’s time dilation, and it’s not just a theory; It’s a fact!

… Now, if you wanted to find out how much slower, you’d have to do some math. We can find D using the Pythagorean theorem

image006

At this point the math get’s really elegant (don’t worry, I won’t show it). Suffice it to say, you plug the D into the Δt’ equation, square the whole expression, collect the Δt’ terms, isolate the Δt’, square root the whole expression, and you end up with this equation

image007

Which is cool, because then we can substitute in Δt to get

image008…the equation for how time in a moving frame relates to time in a rest frame. Boom.

So that’s time travel. Questions are totally appropriate at this time. Stay tuned for the next episode where I’ll show you how to shrink objects.

A time before clocks

13 Mar

Clockwise and counterclockwise are well established terms now, but before clocks were common objects how was rotational motion described?

To describe what we would now call clockwise, the Scottish used the term Sunwise. This relates back to the prior time keeping device, a sundial, and the fact that the Scottish lived in the Northern Hemisphere.

Sundial_2r-1

In the Northern Hemisphere, the sun appears in the southern sky and tracks from East to West. As a result, the shadow on a horizontal sundial tracks from West to East through the North. When clocks replaced sundials, they adopted this traditional sense of rotation.

The Scottish also described clockwise motion with terms related to Deiseil, derived from the Latin dexter, meaning ‘on the right-hand side.’ This is because clockwise motion around an object keeps the right hand toward it (important if you are carrying a sword in said hand).

The Scottish term for counterclockwise was widdershins, related to the German weddersinnes, meaning ‘direction opposite the usual.’ It seems that counterclockwise has again been defined as being the other direction.