Question 1

The mass of a hypothetical planet is 1/100 that of Earth and its radius is 1/4 that of Earth. If a person weighs 600 N on Earth, what would he weigh on this planet?

Accepted Answer

Weight = mass × gravity
The gravity of a planet is given by:
g = G × M / r^2
where G is the gravitational constant, M is the mass of the planet, and r is its radius. We are given that the mass of the planet is 1/100 that of Earth (M' = M/100) and its radius is 1/4 that of Earth (r' = r/4).
So, the gravity on the planet is:
g' = G × M' / r'^2 = G × (M/100) / (r/4)^2 = (G × M / r^2) / 100 = g / 100
Therefore, the person would weigh:
Weight' = mass × g' = mass × (g / 100) = (Weight / 100) = 6 N
So, the answer is (C) 96 N.

Question 2

The speed of a comet in an elliptical orbit about the sun:

Accepted Answer

The answer of The speed of a comet in an...

Question 3

The mass density of a certain planet has spherical symmetry but varies in such a way that the mass inside every spherical surface with center at the center of the planet is proportional to the radius of the surface. If r is the distance from the center of the planet to a point mass inside the planet, the gravitational force on the mass is:

Accepted Answer

The answer of The mass density of a certain planet...

Question 4

Neglecting air resistance, the escape speed from a certain planet for an empty space vehicle is 1.12 *10⁴ m/s. What is the corresponding escape speed for the fully loaded vehicle which has triple the mass of the empty one?

Accepted Answer

The escape speed from a planet is independent of the mass of the escaping object. It only depends on the mass of the planet and the distance from its center.

Question 5

In order to fire a projectile upward and have it escape the Earth's gravity,

Accepted Answer

For a projectile to escape Earth's gravity, its total energy (kinetic plus potential) must not be negative. This means the projectile has enough kinetic energy to overcome the gravitational potential energy of the Earth, allowing it to escape into space without falling back. If the total energy were negative, the projectile would be bound by Earth's gravity.

Question 6

To measure the mass of a planet with the same radius as Earth, an astronaut drops an object from rest (relative to the planet) from an altitude of one radius above the surface. When the object hits its speed is 4 times what it would be if the same experiment were carried out for Earth. In units of M_E (the mass of the Earth), the mass of the planet is:

Accepted Answer

The answer of To measure the mass of a planet...

Question 7

An artificial Earth satellite is moved from a circular orbit with radius R to a circular orbit with radius 2R. During this move:

Accepted Answer

As the satellite moves from a lower orbit to a higher orbit, it gains potential energy. According to the work-energy principle, the work done by a conservative force (like gravity) is equal to the change in potential energy. Since the satellite gains potential energy, the work done by gravity is negative. Therefore, the gravitational force does negative work.

Question 8

A spring scale, calibrated in newtons, is used to weigh sugar. If it were possible to weigh sugar at the following locations, where will the buyer get the most sugar to a newton?

Accepted Answer

The answer of A spring scale, calibrated in newtons, is...

Question 9

A planet travels in an elliptical orbit about a star as shown. At what pair of points is the speed of the planet the same?

Accepted Answer

The answer of A planet travels in an elliptical orbit...

Question 10

Two particles, each of mass m, are a distance d apart. To bring a third particle, with mass 2m, from far away to a resting point midway between the two particles, an external agent must do work equal to:

Accepted Answer

The work done by an external agent to bring a third particle of mass 2m from infinity to a point midway between two particles each of mass m, separated by a distance d, is equal to the negative of the gravitational potential energy at that point. The gravitational potential energy at the midpoint due to the two masses m is given by $U = -\frac{Gm(2m)}{d/2} = -\frac{4Gm^2}{d}$. Since the work done is equal to the change in potential energy, and bringing the mass from infinity (where potential energy is considered zero) to the midpoint increases the potential energy (in a negative sense), the work done is $W = -U = -(-\frac{4Gm^2}{d}) = \frac{4Gm^2}{d}$. However, the correct formula should account for the mass of the third particle being 2m, thus doubling the result: $W = 2 \times \frac{4Gm^2}{d} = \frac{8Gm^2}{d}$, but since we are bringing the mass in against the gravitational pull, the work done is positive in terms of energy input, but when considering the formula for gravitational potential energy, it results in a negative value, hence the correct answer is $-\frac{8Gm^2}{d}$.

Question 11

A rocket ship is coasting toward a planet. Its captain wishes to know the value of g at the surface of the planet. This may be inferred by:

Accepted Answer

The ship's acceleration due to the planet's gravity can be observed and, by using the distance from the planet's center, one can calculate the gravitational acceleration (g) at the planet's surface. This method directly relates to how gravitational force decreases with distance, allowing for the calculation of g at the surface.

Question 12

The escape velocity at the surface of Earth is approximately 11 km/s. What is the mass, in units of M_E (the mass of the Earth), of a planet with twice the radius of Earth for which the escape speed is twice that for Earth?

Accepted Answer

The escape velocity from the surface of a planet of radius R and mass M is given by: $v_e = \sqrt{\frac{2GM}{R}}$ where G is the gravitational constant. For the Earth, $v_e = 11 ext{ km/s}$ and $R = R_\oplus$. For the other planet with radius $2R_\oplus$ and escape velocity $2v_e$, we have: $2v_e = \sqrt{\frac{2GM}{2R_\oplus}} = \sqrt{\frac{GM}{R_\oplus}} \Rightarrow \frac{v_e^2}{GM} = \frac{1}{2R_\oplus}$ Then, the escape velocity is proportional to $\sqrt{M/R}$, so we can write: $\frac{2v_e}{v_e} = \sqrt{\frac{M}{2R_\oplus}} \sqrt{\frac{2R_\oplus}{M}} = \sqrt{2}$ Solving for $M$ gives: $M = 8M_\oplus$ Therefore the answer is (C) 8M_E.

Question 13

A particle might be placed  Rank these situations according to the magnitude of the gravitational force on the particle, least to greatest.

Accepted Answer

The answer of A particle might be placed  Rank...

Question 14

Neglecting air resistance, a 1.0-kg projectile has an escape velocity of about 11 km/s at the surface of Earth. The corresponding escape velocity for a 2.0 kg projectile is:

Accepted Answer

The escape velocity is dependent solely on the mass and radius of the body from which the object is escaping. Therefore, the mass of the projectile does not affect the escape velocity. The escape velocity for a 1.0-kg projectile at the surface of Earth is 11 km/s, so the corresponding escape velocity for a 2.0 kg projectile would also be 11 km/s.

Question 15

A planet travels in an elliptical orbit about a star X as shown. The magnitude of the acceleration of the planet is:

Accepted Answer

The answer of A planet travels in an elliptical orbit...

Question 16

Of the following where would the weight of an object be the least?

Accepted Answer

The weight of an object at the center of the Earth would be the least because all the mass of the Earth would be equally distributed around the object, resulting in a net gravitational force of zero.

Question 17

An astronaut in an orbiting space-craft feels "weightless" because she:

Accepted Answer

An astronaut in an orbiting space-craft is still affected by the force of gravity, but because she and the space-craft are falling towards the Earth at the same rate, she feels weightless. This means that she has the same acceleration as the spacecraft, which is why option D is the correct answer. Option A is incorrect because she is still within the range of Earth's gravity. Option B is incorrect because while there may be a centrifugal force acting on her, it is not the reason she feels weightless. Option C is incorrect because if she had no acceleration, she would be stationary, not in orbit. Option E is incorrect because being outside the Earth's atmosphere has no impact on the astronaut's sensation of weightlessness.

Question 18

An artificial Earth satellite of mass m is moved from a circular orbit with radius R to a circular orbit with radius 2R. If the mass of the Earth is M_E, the work done by the gravitational force is:

Accepted Answer

When an object is moved from one circular orbit to another, the work done by gravitational force can be found using the formula W = ΔU = U_f - U_i, where U is the gravitational potential energy. The potential energy depends on the distance between the satellite and the center of the Earth, so as the distance changes, the potential energy changes. In this case, the initial potential energy of the satellite is -GM_Em/R, since it is in a circular orbit of radius R. The final potential energy is -GM_Em/2R, since it is now in a circular orbit of radius 2R. The difference between the final and initial potential energies is: ΔU = U_f - U_i = (-GM_Em/2R) - (-GM_Em/R) = -GM_Em/R So the work done by the gravitational force is negative, meaning that the force is doing work on the satellite. The best choice is therefore (E).

Question 19

Consider the statement: "Earth moves in a stable orbit around the Sun and is therefore in equilibrium". The statement is:

Accepted Answer

The statement is false because the Earth indeed has a considerable acceleration as it orbits the Sun. This acceleration is centripetal, directed towards the Sun, allowing Earth to maintain its orbit rather than moving in a straight line.

Question 20

A projectile is fired straight upward from Earth's surface with a speed that is half the escape speed. If R is the radius of Earth, the highest altitude reached, measured from the surface, is:

Accepted Answer

The escape speed $v_e$ from Earth is given by the formula $v_e = \sqrt{2gR}$, where $g$ is the acceleration due to gravity and $R$ is the radius of Earth. When a projectile is fired with a speed $v$ that is half the escape speed, $v = \frac{1}{2}v_e = \frac{1}{2}\sqrt{2gR}$, the energy conservation principle can be used to find the maximum altitude $h$ it reaches. At the maximum altitude, the kinetic energy of the projectile is zero, and all the initial kinetic energy has been converted into potential energy.The initial kinetic energy $K_i$ of the projectile is $\frac{1}{2}mv^2$, and the potential energy at the surface of Earth is $-\frac{GmM}{R}$, where $m$ is the mass of the projectile, $M$ is the mass of Earth, and $G$ is the gravitational constant. At the highest point, the potential energy $U_f$ is $-\frac{GmM}{R+h}$, and the kinetic energy is zero.Setting the initial total energy equal to the final total energy gives:$$\frac{1}{2}mv^2 - \frac{GmM}{R} = 0 - \frac{GmM}{R+h}$$Substituting $v = \frac{1}{2}\sqrt{2gR}$ and simplifying, we find that $h = R/3$. This result comes from solving the equation for $h$ in terms of $R$, taking into account that $g = \frac{GM}{R^2}$.

Quiz 13: Gravitation

The mass of a hypothetical planet is 1/100 that of Earth and its radius is 1/4 that of Earth. If a person weighs 600 N on Earth, what would he weigh on this planet?

The speed of a comet in an elliptical orbit about the sun:

Neglecting air resistance, the escape speed from a certain planet for an empty space vehicle is 1.12 *10⁴ m/s. What is the corresponding escape speed for the fully loaded vehicle which has triple the mass of the empty one?

In order to fire a projectile upward and have it escape the Earth's gravity,

An artificial Earth satellite is moved from a circular orbit with radius R to a circular orbit with radius 2R. During this move:

A spring scale, calibrated in newtons, is used to weigh sugar. If it were possible to weigh sugar at the following locations, where will the buyer get the most sugar to a newton?

A planet travels in an elliptical orbit about a star as shown. At what pair of points is the speed of the planet the same?

Two particles, each of mass m, are a distance d apart. To bring a third particle, with mass 2m, from far away to a resting point midway between the two particles, an external agent must do work equal to:

A rocket ship is coasting toward a planet. Its captain wishes to know the value of g at the surface of the planet. This may be inferred by:

The escape velocity at the surface of Earth is approximately 11 km/s. What is the mass, in units of M_E (the mass of the Earth) , of a planet with twice the radius of Earth for which the escape speed is twice that for Earth?

A particle might be placed Rank these situations according to the magnitude of the gravitational force on the particle, least to greatest.

Neglecting air resistance, a 1.0-kg projectile has an escape velocity of about 11 km/s at the surface of Earth. The corresponding escape velocity for a 2.0 kg projectile is:

A planet travels in an elliptical orbit about a star X as shown. The magnitude of the acceleration of the planet is:

Of the following where would the weight of an object be the least?

An astronaut in an orbiting space-craft feels "weightless" because she:

An artificial Earth satellite of mass m is moved from a circular orbit with radius R to a circular orbit with radius 2R. If the mass of the Earth is M_E, the work done by the gravitational force is:

Consider the statement: "Earth moves in a stable orbit around the Sun and is therefore in equilibrium". The statement is:

A projectile is fired straight upward from Earth's surface with a speed that is half the escape speed. If R is the radius of Earth, the highest altitude reached, measured from the surface, is:

Quiz 13: Gravitation

The mass of a hypothetical planet is 1/100 that of Earth and its radius is 1/4 that of Earth. If a person weighs 600 N on Earth, what would he weigh on this planet?

The speed of a comet in an elliptical orbit about the sun:

Neglecting air resistance, the escape speed from a certain planet for an empty space vehicle is 1.12 *104 m/s. What is the corresponding escape speed for the fully loaded vehicle which has triple the mass of the empty one?

In order to fire a projectile upward and have it escape the Earth's gravity,

An artificial Earth satellite is moved from a circular orbit with radius R to a circular orbit with radius 2R. During this move:

A spring scale, calibrated in newtons, is used to weigh sugar. If it were possible to weigh sugar at the following locations, where will the buyer get the most sugar to a newton?

A planet travels in an elliptical orbit about a star as shown. At what pair of points is the speed of the planet the same?

Two particles, each of mass m, are a distance d apart. To bring a third particle, with mass 2m, from far away to a resting point midway between the two particles, an external agent must do work equal to:

A rocket ship is coasting toward a planet. Its captain wishes to know the value of g at the surface of the planet. This may be inferred by:

The escape velocity at the surface of Earth is approximately 11 km/s. What is the mass, in units of ME (the mass of the Earth) , of a planet with twice the radius of Earth for which the escape speed is twice that for Earth?

A particle might be placed Rank these situations according to the magnitude of the gravitational force on the particle, least to greatest.

Neglecting air resistance, a 1.0-kg projectile has an escape velocity of about 11 km/s at the surface of Earth. The corresponding escape velocity for a 2.0 kg projectile is:

A planet travels in an elliptical orbit about a star X as shown. The magnitude of the acceleration of the planet is:

Of the following where would the weight of an object be the least?

An astronaut in an orbiting space-craft feels "weightless" because she:

An artificial Earth satellite of mass m is moved from a circular orbit with radius R to a circular orbit with radius 2R. If the mass of the Earth is ME, the work done by the gravitational force is:

Consider the statement: "Earth moves in a stable orbit around the Sun and is therefore in equilibrium". The statement is:

A projectile is fired straight upward from Earth's surface with a speed that is half the escape speed. If R is the radius of Earth, the highest altitude reached, measured from the surface, is:

Neglecting air resistance, the escape speed from a certain planet for an empty space vehicle is 1.12 *10⁴ m/s. What is the corresponding escape speed for the fully loaded vehicle which has triple the mass of the empty one?

The escape velocity at the surface of Earth is approximately 11 km/s. What is the mass, in units of M_E (the mass of the Earth) , of a planet with twice the radius of Earth for which the escape speed is twice that for Earth?

An artificial Earth satellite of mass m is moved from a circular orbit with radius R to a circular orbit with radius 2R. If the mass of the Earth is M_E, the work done by the gravitational force is: