If Earth were the same size but twice as massive, how would the value of G change, if at all? How would the value of g change, if at all?

Find indefinite integral:

$$\int 6xe^{3x} \,dx$$

If the radius of Earth somehow shrank by half without any change in Earth’s mass, what would be the value of g at the new surface? What would be the value of g above the new surface at a distance equal to the present radius?

Jupiter has about 300 times the mass of Earth and about ten times Earth’s radius. Estimate the size of g on the surface of Jupiter.

What provides the force that causes the centripetal acceleration of a satellite in orbit?

You randomly select an integer from 1 to 100. State whether the events are mutually exclusive. Explain your reasoning. The integer is less than 50; the integer is greater than 40.

The hyperbolic sine function is defined by

$$sinh(x)=(e^x-e^{-x})/2.$$

Prove that sinh is an odd function.

If Earth were twice as massive but remained the same size, what would happen to the value of g?

If the gravitational pull of the moon on Earth were the same over all parts of Earth, would there be any tides? Defend your answer.

Earth and the moon are gravitationally attracted to each other. Does the more massive Earth attract the moon with a greater force, the same force, or less force than the moon attracts Earth?

Suppose you roll a pair of fair, six-sided dice. Let T = the sum of the spots showing on the up-faces. Find the probability distribution of T.

A deuteron is a charged particle that has the same charge as a proton but approximately twice the mass. An alpha particle has twice the charge and approximately four times the mass of a proton. For this problem, assume the mass ratios are exactly 2 and 4. If a proton, a deuteron, and an alpha particle move in a uniform magnetic field in circles having the same radii, compare their (a) speeds, (b) kinetic energies, and (c) angular momentum magnitudes.

A kilogram of water on the side of Earth nearest the moon is gravitationally attracted to the moon with a greater force than a kilogram of water on the side of Earth farthest from the moon. The difference in force per mass, the tidal force, is approximated by

$$F _ { \mathrm { T } } = \frac { 4 G M R } { d ^ { 3 } }$$

where G is the gravitational constant, M is the mass of the moon R is the radius of Earth, and d is the distance between the centers of the moon and Earth. Earth under your feet also stretches you ever so slightly by pulling harder on your feet than on your head. What is Earth’s

$$F_T$$

on you? (Now R is half your height and d is the distance from the center of Earth to you,

$$6.4 \times 10^6 m.$$

M will be Earth’s mass,

$$6.0 \times 10^{24} kg.$$

) How does Earth’s tidal force on you compare with the moon’s?

In a typical romance, the hero's origins are mysterious. He is frequently raised in obscurity before taking his rightful place as leader. How does the story of Arthur reflect this pattern?