Tramplines and Scales and Free Float

[shibbysan]

I'm new here and not sure if this post is in the right spot (it's about Special Relativity) but here goes nothing.

This problem--from the book Spacetime Physics by Taylor and Wheeler--is giving me a lot of trouble.

Fasten a weight-measuring bathroom scale under your feet and bounce up and down on a trampoline while reading the scale. Describe readings on the scale at different times during the bounces. During what part of each jump will the scale have zero reading? Neglecting air resistance, what is the longest part of the cycle during which you might consider yourself to be in a free-float frame?

What I think is that the scale will read zero in the air, because there is no force to be the reciprocal of your mass pushing down. What I don't understand totally is when it is in free float. I understand that free float is trying to isolate a small frame where the effects of gravity can be ignored (special relativity), but how does one explain what is happening with the scale and the trampoline?

 

[pervect]

I'm new here and not sure if this post is in the right spot (it's about Special Relativity) but here goes nothing.

This problem--from the book Spacetime Physics by Taylor and Wheeler--is giving me a lot of trouble.



What I think is that the scale will read zero in the air, because there is no force to be the reciprocal of your mass pushing down. What I don't understand totally is when it is in free float. I understand that free float is trying to isolate a small frame where the effects of gravity can be ignored (special relativity), but how does one explain what is happening with the scale and the trampoline?

This is the correct forum for such questions, welcome to Physics Forums.

It sounds to me like you have answered the original question correctly, though you didn't really fully address the issue of the "longest time" that was originally asked in the textbook.

The "longest time" the scale reads zero consists of your entire trajectory after your feet leave the trampoline, both the upwards and downwards part of the trajectory.

I'm fairly sure that the book will explain (or perhaps just mention) the fact that this "free-fall" part of your trajectory, ignoring air resistance, is a geodesic.

What I don't quite understand is your question. What is it about the scale and the trampoline that you want to explain? What is troubling you?

 

[Zanket]

What I think is that the scale will read zero in the air,

Correct. You're weightless while in the air.

What I don't understand totally is when it is in free float.

It's in free float when you're weightless, while in the air. When you float you are weightless.

I understand that free float is trying to isolate a small frame where the effects of gravity can be ignored (special relativity), but how does one explain what is happening with the scale and the trampoline?

Think of it this way: The only time you're not floating is when you're being pushed. Upon contact of scale to trampoline, the Earth via the trampoline via the scale is pushing on you. Upon release of contact, you're floating. Whenever you're floating, your frame is a free-float frame and you're weightless.

As you alluded, a free-float frame is technically infinitesimally small, to make negligible the tidal force within it. A better way to think of it is: a free-float frame is a frame of any size, in free fall, throughout which the tidal force is negligible. In this way a free-float frame is practically identical to the any-size inertial frame of special relativity, the only difference being a negligible tidal force vs. no tidal force. The name "free-float" is redundant to me, so I think of it as "free to float".

 

[shibbysan]

So even though gravity is affecting you you're in free float for the time in the air (in regards to the "frame" around you)? Obvsiouly gravity is still acting on you because you come back down, but the trick is then to establish a frame around you where--relative to the frame--gravity does not affect you?

 

[Zanket]

So even though gravity is affecting you you're in free float for the time in the air (in regards to the "frame" around you)?

Yes (but a better viewpoint regarding gravity affecting you is below). While in the air you are floating and the frame around you (that is, an imaginary box surrounding you with respect to which you are at rest) is a free-float frame.

Obvsiouly gravity is still acting on you because you come back down, but the trick is then to establish a frame around you where--relative to the frame--gravity does not affect you?

You got it. You say “gravity is still acting on you” yet contradictorily “gravity does not affect you”. Let’s clean that up. You pushed off the trampoline to move away from the Earth, took no further action (began floating), and the Earth met you again. Then the Earth must have moved to reclaim you. When you want to go somewhere by car, you must initially accelerate to push the car there. And sure enough, while planted on the Earth we feel an upward acceleration, a push. How can the Earth push upward in all directions and not expand in size? Imagine that space itself accelerates inward toward the center of the Earth, while the surface of the Earth stays in place. Now the surface is accelerating through space like your car accelerates through space, but the Earth does not expand. With this viewpoint, gravity acts on you/affects you while you float no more than does some passing accelerating rocket, which is to say it does not. Since gravity does not affect your frame, special relativity applies within. You’ll eventually be reclaimed by the Earth because the space in which you float is going in that direction. But space is moving inward only so fast at each altitude, the higher the altitude the lesser inward velocity, so if you move outward fast enough (escape velocity) you’ll escape meeting the Earth, like coasting on a bicycle against a decelerating walkway.

With this viewpoint, gravity affects and acts on the space (or technically spacetime) in which you float, but not you per se. In general relativity a key idea is: Spacetime tells mass how to move; mass tells spacetime how to curve. To paraphrase: The spacetime in which you float moves you toward the Earth; the Earth tells spacetime how to move toward the Earth. When you accelerate because the spacetime in which you float accelerates, you do not feel an acceleration; this is gravitational acceleration. A caveat is the tidal force mentioned above.

 

[pervect]

So even though gravity is affecting you you're in free float for the time in the air (in regards to the "frame" around you)? Obvsiouly gravity is still acting on you because you come back down, but the trick is then to establish a frame around you where--relative to the frame--gravity does not affect you?

Let's go from a trampoline to a more ambitious implementation of the same idea. Astronauts-in-training, generally curious peopole, and movie-makers have all made use of the "Vomit Comit", a plane that follows a free-fall parabolic trajctory.

Like the person on the trampoline while he is in the air, people in the plane experience weightlessness.

You can imagine people in the plane carrying out physics experiments. They would set up a "frame" (probably it would be better to call it a coordinate system, because the word "frame" has a lot of extra baggage that doesn't necessarily apply) with which they could do measurements. The people in the plane would find that Newton's laws hold (well, there would be some very tiny effects due to tidal forces). In this coordiante system, there would be no gravity (except for the very small tidal forces I mentioned). The equations of motions in this coordiante system would be just

m x'' = fx, my'' = fy, mz'' = fz, where the double-prime represents the second derivative, i.e. x'' = d^2x/dt^2.