Mastering Astronomy Chapter 4

increase

a rock starting from rest on a high cliff, then moving faster and faster as it falls (the law of conservation of energy tells us that energy is conserved as the rock falls, so as it loses gravitational potential energy (by losing altitude) it gains kinetic energy (by increasing its speed)

hyperbolic parabolic elliptical

the gravitational force depends only on the product of the masses

the gravitational force exerted on the asteroid on the left by the asteroid on the right will be equal in magnitude, but opposite in direction to the gravitational force exerted on the object on the right by the left

the asteroid with the largest acceleration will be the one that has the strongest gravitational force exerted on it by the object on the right - the Moon has a larger acceleration than Earth, because it has a smaller mass (F=ma, for a particular force F, the product mass x acceleration must always be the same. therefore, if mass is larger, acceleration must be smaller and vice versa)

the strength of gravity follows an inverse square law with distance

its total angular momentum is conserved

in the absence of external influences (torques), the angular momentum of an object of a system of objects stays constant (since the angular momentum of an object depends on both its size and rate of rotation, the cloud's rate of rotation will increase as its size (or radius) decreases in order to conserve angular momentum)

gravitational potential energy being converted to thermal energy (as the cloud shrinks in size, its gravitational potential energy decreases. Because energy cannot simply disappear, the "lost" gravitational potential energy must be converted into some other form. Some of it is converted into thermal energy, which raises the temperature of the gas cloud. The rest is mostly converted into radiative energy, which is released into space as light)

A tremendous amount of gravitational potential energy would be converted into other forms of energy, and the Sun would spin much more rapidly. (the dramatic shrinkage of the Sun would mean the loss of a huge amount of gravitational potential energy. Because energy is always conserved, this "lost" gravitational potential energy must reappear in other forms, such as heat (thermal energy) and light (radiative energy). Meanwhile, conservation of angular momentum would ensure that the collapsed Sun would spin much faster)

It will end up on an orbit that is farther from the Sun (total energy must be conserved, so if one asteroid loses energy and moves to a closer orbit, the other must gain energy and move to a more distant orbit)

in the absence of air resistance, all objects on Earth should fall with the same acceleration of gravity regardless of mass

Scientists dropping balls on the Moon find that balls of different mass fall at slightly different rates (dropping the balls on the Moon removes any potential effects due to air resistance, so a result in which mass affects the rate of fall would directly contradict the prediction of Einstein's (as well as Newton's) theory)

One faces the Moon and one faces opposite the Moon (the tidal bulges face toward and away from the Moon, because they are caused primarily by the gravitational attraction between Earth and the Moon. Friction explains why the bulges are slightly ahead of the Earth-Moon line, rather than directly on the Earth-Moon line. We'll ignore that detail for now)

Yes, though land rises and falls by a much smaller amount than the oceans.

two high tides and two low tides each day

The second tidal bulge arises because gravity weakens with distance, essentially stretching Earth along the Earth-Moon line.

- low tides are lowest at both full moon and new moon - high tides as highest at both full moon and new moon (those are the times when the tidal forces of the Sun and Moon align (and therefore add to one another). Therefore, high tides are higher and low tides are lower at these times, which are called spring tides. (In contrast, we have neap tides at first- and third-quarter moons, when high tides are not as high and low tides are not as low.)

times when the tidal forces of the Sun and Moon align (and therefore add to one another). therefore, high tides are higher and low tides are lower at these times

at first and third quarter moon, when high tides are not as high and low tides are not as low

because the gravitational attraction between Earth and the Moon varies more across Earth than does the gravitational attraction between Earth and the Sun (the Sun exerts a stronger gravitational force on Earth, which is why Earth orbits the Sun. However, tides are caused by the variation in the gravitational attraction across Earth. Even though the gravitational attraction between Earth and the Moon is smaller than the attraction between Earth and the Sun, the Moon's much closer distance makes this attraction vary more across Earth. That is why tides are due primarily to the Moon, with only a secondary effect from the Sun)

it is going around a circular track at a steady 100 mph

mass would be the same but your weight would be different

gradually transforms into other forms of energy

Earth's orbit would not change

greater than your normal weight at rest (increasing speed means acceleration, and when the elevator is accelerating upward you will feel a force pressing you to the floor, making your weight greater than your normal (at rest) weight)

less than your normal weight at rest (even though the elevator is still moving upward, the fact that its speed is slowing means that the acceleration is downward. The situation is rather like that of a ball that is still on its way up after you throw it: the ball slows as it goes upward because of the downward acceleration of gravity. Because the acceleration of the elevator is downward, your weight is lower than normal)

the elevator moves downward while slowing in speed

the force you exert on the scale - your presence in an elevator cannot change either your mass or the gravitational force exerted on you by Earth

You float weightlessly within the elevator car because you and the elevator both begin to accelerate downward at the same rate (Once the cable breaks, you and the elevator car both fall with the acceleration of gravity. This means you are no longer pressing against the scale or the elevator floor, so you float weightlessly within the car -- though only until you and the car hit the ground!)

one-quarter as strong