so picture this: i have a yo-yo which i am swinging around in the air. the string is in my hand, around which the yo-yo is describing a circle. consider this a model of the earth around the sun.
now, said yo-yo begins to rotate faster, and the tension across the string becomes too great. the string snaps, and the yo-yo flies across my backyard and whonks the weird neighbor kid in the head.
extend this analogy. what the hell would be breaking? space? can you really break space? or maybe gravity. boy would there be egg on my face- ‘sorry guys, i broke gravity.’
i tried mentally modeling the problem, but that’s what got me in this mess in the first place. the only thing i can think of is that space (or gravity, or whatever) is a lot more stretchable than string, and that the earth would simply describe a bigger and bigger circle. so that the point of breakage would be when the velocity of the earth reached light speed.
can anybody give me a hand here? i wanna have a clear mind when i go to sleep tonight. certainly don’t wanna be dreaming about breaking physical laws (‘Oh my god! Planck’s constant just up and disappeared!’)
The string analogy isn’t a very good one so you’ll just frustrate yourself trying to use it at a model.
A better analogy are those curved funnel thingies that you can roll coins into at the local museum. The curved walls of the funnel are a good model of gravity, decreasing with the distance from the center.
I won’t go into orbital mechanics but the attractive force of gravity decreases with the inverse square of the distance from the center. If you double the distance, the relative force or gravity is a reduced to a quarter of the first distance. To maintain a stable orbit the outward force has to be equal to the force of gravity. That’s why high orbits like communcation satellits are 24 hours and low orbers are an hour or two.
See the physics section of your local library for more details.
The string model will give you a headache eventually. Abandon it. IF the yo-yo flew out through space it would not increase in velocity due to the escape. Of course, something caused it to escape but it wasn’t a string breaking. It was because it fell off the stack of turtles that were holding it up (old joke). It would continue at the escape speed until it was captured by another body, or crashed into something. It would never approach the speed of light…unless it fell into a black hole…and it would not tear the fabric of the universe. Imagine it as rolling across a table top and pushing an indentation in the tablecloth under itself.
Do the thought experiment this way: You are a satellite in orbit 35,800 kilometers above Manhattan. You are continually falling toward the earth, but the earth is falling away from you at the same speed you are falling. You are in a geostationary orbit. If you fire a thruster and shoot off into space, you will not continue to increase your speed until you reach the speed of light. In order to continue acceleration, you must provide a continuing source of propulsion.
Like the others said, the string analogy is misleading you. But, trying to extend your analogy as you asked, you wouldn’t be “breaking” space or gravity, you would be suddenly removing the gravity that is holding Earth in its orbit (like, if the sun were to magically vanish from existance). Without the sun’s gravity pulling the Earth into a nice orbit, the Earth would continue off on a tangent. (like some threads when a moderator vanishes :D)
i am swinging this yo-yo around, and i make it go faster and faster. as the speed increases, so does the radius of the circle described (albeit only slightly). at some point, the rope breaks and the yo-yo goes off flying. neighbor kid gets bonked.
if i were to increase the velocity of some body orbiting a larger body, the orbital radius would increase. in the earlier example, the string is providing the centripetal acceleration; in the latter, gravity. now we know that the string can snap. can the velocity of the orbiting body increase to the point that the provider of centripetal acceleration ceases to do so?
and i realize that the string analogy must be flawed somehow. could anybody point out why exactly?
Basically, what you’re describing is the concept of escape velocity. As the orbital velocity increases, the radius also increases. If the velocity is high enough, the object is no longer in orbit, it is travelling freely through space.
The reason the string analogy doesn’t work is that the break is a discontinuity. The force disappears all at once. With gravity, the force cannot disappear, since it is inherent in the shape of space itself. Instead, it gets less and less until it is no longer a significant part of the motion equation for the object.
Imagine this. If your yo-yo string were to stretch more and more until it was adding several feet per second. It would no longer be affecting the movement of the yo-yo at that point.
While Saltire’s answer about why your analogy is flawed is correct, I’d like to add two things.
First, the Force that the string exerts on the yo-yo does not behave the same as the Force that the Sun exerts on a planetoid. Gravity follows the Inverse Square Law, as Padeye so punctually pointed out. If you increase the distance between the two bodies by a factor of w, the Force between them will decrease by a factor of w². The string Force follows Hooke’s Law (or at least something similar to it), which is the Force Law governing springs. In this case, when you increase the distance between your hand and the yo-yo, the Force between them will likewise increase. This distinction has many implications, one of which I’ll explain now.
Second, you said, “if i were to increase the velocity of some body orbiting a larger body, the orbital radius would increase.” While this is technically correct, it is misleading. If you were to increase a planetoid’s speed around the Sun, it would go into a higher orbit. However, in achieving that orbit, it would lose some speed. More speed, even, than what you gave it. So, when a body goes to a higher orbit, it does, in fact, slow down.
My badly researched theory is that your analogy is the reason that people for centuries assumed that orbits were circular. You see circular orbits in things like yo-yos, but as far as the Ancients knew, elliptical orbits did not occur in nature. The simple truth of it, though, is that planetoids do not behave like yo-yos, unless you count the Moon’s going “Around the World”.