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A science-fiction tale describes an artificial "planet" in theform of a band completely encircling a sun (Fig. $31 ) .$ Theinhabitants live on the inside surface (where it is alwaysnoon). Imagine that this sun is exactly like our own, that thedistance to the band is the same as the Earth-Sun distance(to make the climate temperate), and that the ring rotatesquickly enough to produce an apparent gravity of $g$ as onEarth. What will be the period of revolution, this planet'syear, in Earth days?
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Physics 101 Mechanics
Gravitation and Newton's Synthesis
Newton's Laws of Motion
Applying Newton's Laws
University of Washington
Simon Fraser University
In physics, dynamics is the branch of physics concerned with the study of forces and their effect on matter, commonly in the context of motion. In everyday usage, "dynamics" usually refers to a set of laws that describe the motion of bodies under the action of a system of forces. The motion of a body is described by its position and its velocity as the time value varies. The science of dynamics can be subdivided into, Dynamics of a rigid body, which deals with the motion of a rigid body in the frame of reference where it is considered to be a rigid body. Dynamics of a continuum, which deals with the motion of a continuous system, in the frame of reference where the system is considered to be a continuum.
In physics, orbital motion is the motion of an object around another object, which is often a star or planet. Orbital motion is affected by the gravity of the central object, as well as by the resistance of deep space (which is negligible at the distances of most orbits in the Solar System).
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6.69. So we're told about some science fiction story that ER imagines an artificial planet that's made of is this sort of band in circling a star. People live on the inside surface of it. It's always noon, which is maybe a bit of a problem. We want to imagine that this is the star is the same as our own son and that the distance from the band to the star is the same as the Earth's sun distance. So one astronomical unit and that the ring is rotating quickly enough that it produces an apparent gravity of gene That's the same as on Earth. We want to figure out what its period of revolution is, an earth beings. So saying that we want an apparent gravity equal to that of Earth of the Surface of the Earth is the same as saying that the normal force experienced by someone, uh, you know, on the inside of this has to be the same as, uh as G. This will be the forest for gravity. They're both pointing towards our star here. And then, of course, this has to be. These have to add up to be equal to the centripetal acceleration, which is going to be m v squared over r So writing down, um, you know that we have the force of gravity. Plus the normal force has to be m times he squared over r. And so you know something Moving around in a circle like constant speed, its speed has to be the distance that it travels circumference of the circle provided by the amount of time it takes to travel that distance. We want, uh, efs a band to be equal to m times g of Earth and the force from the star This point to be g times the mass of the object times the mass of the star provided by r squared. So you just put these in and solve for the, uh, a period? We, um we have mm times four pi squared. Well, heaven r squared over R. So what are over? T square is equal to m G plus G and s over r squared. So the mass of our hypothetical person or whatever on the surface, inside surface of this ribbon band, whatever it doesn't matter, which is good would sort of be an indication that we've done something wrong. If you didn't cancel out, so then the period, it's going to be the square root. The four pi squared R over G times the mass of this star divided by r squared, plus our 9.8 meters per second squared that we want the grab apparent gravity to be, and so that this works out to be 8.99 days. And if you go on, work out what this is because the earth's sun distances so large, even though the mass of the sun is also large. But the distance is large and that G is very small, so it ends up. It turns out you could actually ignore this in the calculations without really changing this whole lot. It's sort of an interesting thing. You could just have this disk out spinning, you know, in space nowhere near anything else. And it would still need about a nine day rotational period to produce an apparent gravity, the same as the Earth's surface
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