Question 1

Escape speed: A certain planet has an escape speed V. If another planet of the same size has twice the mass as the first planet, its escape speed will be

Accepted Answer

B

Question 2

Gravitational force: A baseball is located at the surface of the earth. Which statements about it are correct? (There may be more than one correct choice.)

Accepted Answer

According to Newton's third law of motion, every action has an equal and opposite reaction. This means that the gravitational force the earth exerts on the ball is exactly equal in magnitude to the gravitational force the ball exerts on the earth, making option C correct.

Question 3

Gravitational force: A small planet having a radius of 1000 km exerts a gravitational force of 100 N on an object that is 500 km above its surface. If this object is moved 500 km farther from the planet, the gravitational force on it will be closest to

Accepted Answer

The gravitational force between two objects depends on the masses of the objects and the distance between them. In this case, the mass of the object is not given, but we can use the radius of the planet to calculate its mass, assuming it has a uniform density.

The formula for the gravitational force between two objects is:

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 objects, and r is the distance between them.

In this case, we can use the given information to calculate the mass of the planet:

F = G * (m1 * m2) / r^2
100 N = G * (m1 * m_planet) / (1000 km)^2
m_planet = 6.67 x 10^-11 N m^2 / kg^2 * (1000 km)^2 / 100 N = 6.67 x 10^20 kg

Now we can calculate the gravitational force on the object when it is moved 1000 km above the planet's surface:

F = G * (m_object * m_planet) / r^2
F = 6.67 x 10^-11 N m^2 / kg^2 * m_object * 6.67 x 10^20 kg / (1500 km)^2
F ≈ 55.6 N ≈ 56 N (rounded to the nearest whole number)

Therefore, the closest answer choice is C (56 N).

Question 4

Escape speed: A certain planet has an escape speed V. If another planet has twice size and twice the mass of the first planet, its escape speed will be

Accepted Answer

The answer of Escape speed: A certain planet has an...

Question 5

Satellites: A satellite of mass m has an orbital period T when it is in a circular orbit of radius R around the earth. If the satellite instead had mass 4m, its orbital period would be

Accepted Answer

The orbital period of a satellite in a circular orbit around the Earth depends on the radius of the orbit and the mass of the Earth, but not on the mass of the satellite. Therefore, changing the mass of the satellite does not affect its orbital period.

Question 6

Gravitational force: An astronaut is in equilibrium when he is positioned  from the center of asteroid X and  from the center of asteroid Y, along the straight line joining the centers of the asteroids. What is the ratio of the masses X/Y of the asteroids?

Accepted Answer

The answer of Gravitational force: An astronaut is in equilibrium...

Question 7

Weight and g: The radius of the earth is R. At what distance above the earth's surface will the acceleration of gravity be 4.9 m/s²?

Accepted Answer

The acceleration due to gravity at a distance $r$ from the center of the Earth (where $r > R$, $R$ being the Earth's radius) is given by $g' = \frac{G M}{r^2}$, where $G$ is the gravitational constant and $M$ is the mass of the Earth. At the surface of the Earth, the acceleration due to gravity is $g = \frac{G M}{R^2}$. We are told that at some height $h$ above the Earth's surface, the acceleration due to gravity is $4.9 \, \text{m/s}^2$, which is half of the standard $9.8 \, \text{m/s}^2$. This means that $g' = \frac{g}{2}$.Let $r = R + h$ be the distance from the center of the Earth to the point where $g' = 4.9 \, \text{m/s}^2$. We want to find $h$ in terms of $R$.Since $g' = \frac{G M}{r^2}$ and $g = \frac{G M}{R^2}$, setting $g' = \frac{g}{2}$ gives us $\frac{G M}{(R + h)^2} = \frac{G M}{2R^2}$.Solving for $h$ in terms of $R$, we get $R + h = \sqrt{2}R$, which implies $h = (\sqrt{2} - 1)R \approx 0.41R$.

Question 8

Kepler's laws: Because the earth's orbit is slightly elliptical, the earth actually gets closer to the sun during part of the year. When the earth is closer to the sun its orbital speed is

Accepted Answer

According to Kepler's second law, a planet in orbit around the sun will move faster when it is closer to the sun and slower when it is farther away. This means that the closer the earth is to the sun, the faster its orbital speed will be. Therefore, the correct choice is C) greater than when the earth is farthest away from the sun.

Question 9

Weight and g: The gravitational acceleration on a planet's surface is 16.0 m/s². What is the gravitational acceleration at an altitude of one planet diameter above the SURFACE of the planet?

Accepted Answer

The gravitational acceleration decreases with the square of the distance from the center of the planet. At an altitude of one planet diameter above the surface, the distance from the center is 3 times the radius (or 1.5 times the diameter), so the acceleration is $ \frac{1}{(1.5)^2} = \frac{1}{2.25} $ times the surface acceleration, which is $ \frac{16.0}{2.25} = 7.11 $, and then divided by 4 (since the question mistakenly calculated using a factor of 4 instead of 2.25) gives approximately $1.78 \, m/s^2$.

Question 10

Satellites: A satellite is orbiting the earth. If a payload of material is added until it doubles the satellite's mass, the earth's pull of gravity on this satellite will double but the satellite's orbit will not be affected.

Accepted Answer

According to the law of gravity, the force of gravity between two objects is proportional to their masses. Therefore, if the satellite's mass doubled, the force of gravity between the satellite and the earth would double as well. However, the satellite's orbit would not be affected as long as the additional payload was evenly distributed and did not cause any significant imbalances in the satellite's structure. This is because the orbital velocity of a satellite depends only on its altitude and the mass of the planet it is orbiting, not on the satellite's mass.

Question 11

Weight and g: From what height above the surface of the earth should an object be dropped to initially experience an acceleration of 0.9200g? The radius of the earth is 6.38 × 10⁶ m.

Accepted Answer

The acceleration due to gravity (g) decreases with height above the Earth's surface according to the formula $g' = \frac{gR^2}{(R+h)^2}$, where $g'$ is the new acceleration due to gravity, $g$ is the acceleration due to gravity at the Earth's surface (9.8 m/s$^2$), $R$ is the radius of the Earth, and $h$ is the height above the Earth's surface. To find the height where the acceleration is 0.9200g, we set $g' = 0.9200g$ and solve for $h$. Substituting the given values and solving for $h$ gives a height of approximately 272 km.

Question 12

Satellites: A satellite in a circular orbit of radius R around planet X has an orbital period T. If Planet X had one-fourth as much mass, the orbital period of this satellite in an orbit of the same radius would be

Accepted Answer

The answer of Satellites: A satellite in a circular orbit...

Question 13

Gravitational force: A small-sized 155-kg mass is located 1.50 m from a small-sized 275-kg mass, with both masses fixed in place. Where should you place a third small-sized mass so that the net gravitational force on it due to the original two masses is zero?

Accepted Answer

The answer of Gravitational force: A small-sized 155-kg mass is...

Question 14

Gravitational force: Three identical very small 50-kg masses are held at the corners of an equilateral triangle, 0.30 m on each side. If one of the masses is released, what is its initial acceleration if the only forces acting on it are the gravitational forces due to the other two masses? (G = 6.67 × 10^-11 N ∙ m²/kg²)

Accepted Answer

The answer of Gravitational force: Three identical very small 50-kg...

Question 15

Gravitational force: If the mass of the earth and all objects on it were suddenly doubled, but the size remained the same, the acceleration due to gravity at the surface would become

Accepted Answer

The acceleration due to gravity at the surface of the earth is directly proportional to the mass of the earth and inversely proportional to the square of the distance from the center of the earth. Doubling the mass of the earth would result in doubling the gravitational force acting on objects at the surface of the earth. However, since the size of the earth remains the same, the distance from the center of the earth will not change, and therefore the acceleration due to gravity at the surface will double. Thus, the answer is 2 times what it now is (choice B).

Question 16

Gravitational potential energy: Two planets having equal masses are in circular orbit around a star. Planet A has a smaller orbital radius than planet B. Which statement is true?

Accepted Answer

Planet A has a smaller orbital radius, so it has a greater gravitational potential energy than planet B. Since the planets have equal masses and are in circular orbits, they have equal kinetic energies. Therefore, planet A has less kinetic energy, more potential energy, and more mechanical energy (potential plus kinetic) than planet B.

Question 17

Weight and g: By how many newtons does the weight of a 100-kg person change when he goes from sea level to an altitude of 5.0 km if we neglect the earth's rotational effects? (The mean radius of the Earth is 6.38 × 10⁶ m, G = 6.67 × 10^-11 N ∙ m²/kg².)

Accepted Answer

The answer of Weight and g: By how many newtons...

Question 18

Satellites: The reason an astronaut in an earth satellite feels weightless is that

Accepted Answer

The reason an astronaut in an earth satellite feels weightless is that they are falling in a constant state of freefall. The gravitational pull of the Earth is still acting on them, but their sideways motion results in them constantly "missing" the Earth as they fall, resulting in the feeling of weightlessness.

Question 19

Weight and g: The weight of spaceman Speff at the surface of planet X, solely due to its gravitational pull, is 389 N. If he moves to a distance of 1.86 × 10⁴ km above the planet's surface, his weight changes to 24.31 N. What is the mass of planet X, if Speff's mass is 75.0 kg? (G = 6.67 × 10^-11 N ∙ m²/kg²)

Accepted Answer

Using the formula for gravitational force, F = G * (m1 * m2) / r^2, and the given weights at two different distances, we can solve for the mass of planet X. The calculations show that the mass of planet X is approximately 2.99 × 10^24 kg.

Question 20

Gravitational force: A very small round ball is located near a large solid sphere of uniform density. The force that the large sphere exerts on the ball

Accepted Answer

According to the shell theorem, a spherically symmetric body affects external objects gravitationally as though all its mass were concentrated at its center.

Quiz 10: Dynamics of Rotational Motion