1. If our sun shrank in size to become a black hole, using the gravitational force equation, explain why the Earth's orbit would not be affected.

2. Strictly speaking, you weigh a tiny bit less when you are in the lobby of a massive skyscraper than you do at home. Why is this so?
3. Ultimately, the universe may expand without limit, or it may coast to a stop, or it may turn around and collapse to a "big crunch." What is the single most important quantity that will determine which of these fates is in store for the universe?
4. Which requires more fuel - a rocket going from the Earth to the moon or a rocket coming from the moon to the Earth?
I know more fuel is required to go from the earth to the moon but why?

1. The "gravitational force equation" depends only upon the mass of the central body and the distance from it. The size of the body makes no difference.

2. There is a lot of mass above you pulling you away fromthe center of the Earth.

3. The rate of expansion of the universe (Hubble constant) and the density are equally important. See http://en.wikipedia.org/wiki/Critical_density
The apparent acceleration of the rate of expansion is not understood, not are the roles of dark energy and dark matter. We really cannot make predictions at this time.
4. Getting to the moon requires much more energy for two reasons:
(a) Going there, you need to take extra supplies necesary for life support and getting back again.
(b) The atmosphere of the Earth provides most of the deceleration necessary to land on earth safely, with heat shields, parachutes or lifting-body airplane shapes like Space Shuttle.

1. If our sun shrank in size to become a black hole, the Earth's orbit would not be affected due to the conservation of angular momentum. Angular momentum is a property of rotating objects and is determined by the product of an object's moment of inertia and its angular velocity. In the case of the Earth's orbit, it depends on the mass of the Sun, the distance between the Sun and the Earth, and the Earth's orbital speed.

When the Sun becomes a black hole, its mass would be compressed into a much smaller volume, resulting in a significantly higher density. However, its total mass would remain unchanged. The gravitational force between the Earth and the Sun is determined by Newton's law of gravitation, which states that the force is proportional to the product of the masses and inversely proportional to the square of the distance between them.

Since the mass of the Sun (now a black hole) remains the same, and the distance between the Earth and the black hole doesn't change significantly, the gravitational force experienced by the Earth would also remain relatively constant. As a result, the Earth's orbit would not be disturbed as long as the mass of the black hole remains the same.

2. When you are in the lobby of a massive skyscraper, you would weigh a tiny bit less compared to being at home due to the effect of gravity. This is because the gravitational force you experience is slightly weaker at higher altitudes.

Gravity decreases with distance from the center of the Earth, so the further you are from the Earth's center, the weaker the gravitational pull. In a massive skyscraper, you are slightly farther away from the center of the Earth compared to when you are at home. As a result, the gravitational force acting on you is slightly reduced, causing a very small decrease in weight.

This decrease in weight due to the change in altitude within a skyscraper is barely noticeable on a human scale. It would require sophisticated measuring equipment to detect such a minute difference. Nonetheless, strictly speaking, your weight would be slightly lower in a skyscraper lobby compared to being at home.

3. The single most important quantity that will determine the fate of the universe (whether it will expand without limit, coast to a stop, or collapse to a "big crunch") is the density parameter, often denoted as "Omega" (Ω).

Omega is a dimensionless number that compares the actual density of matter and energy in the universe to the critical density required for the universe to have a particular fate. If Omega is greater than 1, the universe is considered to be "closed" and will eventually collapse in a "big crunch." If Omega is equal to 1, the universe is "flat" and will continue to expand but at a gradually decreasing rate. If Omega is less than 1, the universe is "open" and will expand forever, with the expansion gradually slowing down.

The precise value of Omega determines which of these scenarios will occur, and it depends on the combination of matter, dark matter, dark energy, and their respective densities. Measurements and observations from cosmology help scientists estimate Omega and understand the long-term fate of the universe.

4. More fuel is required for a rocket to go from the Earth to the moon compared to a rocket coming from the moon to the Earth due to the difference in gravitational potential energy and the need to overcome Earth's gravity.

When a rocket takes off from the Earth's surface, it needs to overcome the Earth's gravity, which requires a significant amount of energy. This energy is provided by the fuel burned by the rocket engines. Additionally, during the ascent, the rocket needs to reach escape velocity to break free from Earth's gravitational pull.

On the other hand, when a rocket is coming from the moon to the Earth, it benefits from the moon's lower gravity. The moon has about one-sixth the gravity of Earth, which means the rocket requires less energy to launch and reach escape velocity. As a result, less fuel is needed for a rocket to return from the moon to the Earth compared to going from the Earth to the moon.

In summary, the higher amount of fuel required to go from the Earth to the moon is primarily due to the need to overcome Earth's stronger gravity and reach escape velocity.

1. If our sun were to shrink in size to become a black hole, the Earth's orbit would not be affected significantly. This is because the gravitational force equation shows that the gravitational force between two objects depends on their masses and the distance between them. The equation is given as F = G * (m1 * m2) / r^2, where F is the gravitational force, G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between their centers.

While the mass of the Sun would change if it were to become a black hole, the distance between the Sun and the Earth would remain relatively the same. The Earth's orbit is determined by the balance between the gravitational force from the Sun and its tangential velocity. Since the distance between the Earth and the Sun would remain relatively constant, the gravitational force between them would not change significantly, and therefore, the Earth's orbit would not be significantly affected.

2. When you are inside a massive skyscraper, you weigh slightly less compared to when you are at home due to the effect of gravity. This is because the gravitational force on you is slightly weaker inside the skyscraper compared to outside.

Gravity is stronger closer to the center of the Earth and weaker as you move further away. When you are in a skyscraper, you are slightly further away from the center of the Earth compared to when you are at home, hence experiencing slightly weaker gravity. The effect is very small because buildings are not generally tall enough to cause a significant difference in gravity, but it can be measured with sensitive instruments.

3. The single most important quantity that will determine the fate of the universe is its density. The density of matter and energy in the universe will determine whether it expands indefinitely, coasts to a stop, or collapses in a big crunch.

If the density of matter and energy is below a critical value, then the universe will continue to expand indefinitely. This is known as an open or accelerating universe.

If the density is exactly at the critical value, the universe will coast to a stop after a certain amount of time. This is called a flat universe.

If the density is above the critical value, the universe will eventually stop expanding and start to contract, leading to a big crunch where everything collapses back into a singularity.

Currently, the evidence suggests that the universe's density is close to the critical value, making its ultimate fate uncertain.

4. More fuel is required for a rocket going from Earth to the moon compared to a rocket coming from the moon to Earth because of the differences in gravity and the need to overcome Earth's gravitational pull.

When a rocket launches from Earth, it needs to overcome Earth's gravitational force, which requires a significant amount of energy. This energy comes from the rocket's fuel, and the rocket needs to carry enough fuel to generate sufficient thrust to overcome Earth's gravity in order to reach the moon.

On the other hand, when a rocket leaves the moon and heads towards Earth, it is already outside Earth's gravitational pull. The moon's gravity is much weaker than Earth's, so the rocket requires less fuel to escape the moon's gravity and begin its journey towards Earth.

In summary, more fuel is needed to go from Earth to the moon due to the higher gravitational force of Earth, whereas less fuel is needed to come back from the moon to Earth because the moon's gravity is weaker.