Question 1

A spherical shell has inner radius R₁, outer radius R₂, and mass M, distributed uniformly throughout the shell.The magnitude of the gravitational force exerted on the shell by a point mass m a distance d from the center, inside the inner radius, is:

Accepted Answer

The answer of A spherical shell has inner radius R₁,...

Question 2

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

The gravitational force is proportional to the mass of the planet and inversely proportional to the square of its radius. The weight on the new planet is (1/100) of Earth's mass divided by (1/4)^2 of Earth's radius, which simplifies to 1/16. Therefore, 600 N * 1/16 = 37.5 N. However, the correct calculation should be 600 N * (1/100) / (1/16) = 96 N.

Question 3

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 gravitational force is weakest near the center of Earth, so the same force (1 newton) would correspond to a larger mass of sugar compared to other locations.

Question 4

A spherical shell has inner radius R₁, outer radius R₂, and mass M, distributed uniformly throughout the shell.The magnitude of the gravitational force exerted on the shell by a point particle of mass m, located a distance d from the center, outside the inner radius and inside the outer radius, is:

Accepted Answer

The answer of A spherical shell has inner radius R₁,...

Question 5

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 6

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 $g = \frac{GM}{R^2}$.

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

The answer of An artificial Earth satellite is moved from...

Question 8

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 9

Mars has a mass of about 0.1075 times the mass of Earth and a diameter of about 0.533 times the diameter of Earth.The acceleration of a body falling near the surface of Mars is about:

Accepted Answer

The acceleration due to gravity on a planet's surface is proportional to its mass and inversely proportional to the square of its radius. Given Mars' mass and diameter relative to Earth, the acceleration is approximately 3.7 m/s².

Question 10

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 11

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 12

Each of the four corners of a square with edge a is occupied by a point mass m.There is a fifth mass, also m, at the center of the square.To remove the mass from the center to a point far away the work that must be done by an external agent is given by:

Accepted Answer

The answer of Each of the four corners of a...

Question 13

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 for an object only depends on the mass of the planet and the distance from the center of the planet. It does not depend on the mass of the projectile. Therefore, the escape velocity for a 2.0 kg projectile would be the same as that for a 1.0 kg projectile, which is 11 km/s.

Question 14

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 15

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 $v_e$ from a planet is given by the formula $v_e = \sqrt{\frac{2GM}{r}}$, where $G$ is the gravitational constant, $M$ is the mass of the planet, and $r$ is the radius of the planet. If the escape velocity is doubled ($2v_e$) and the radius is also doubled ($2r$), we can set up the equation $\sqrt{\frac{2G(2M)}{2r}} = 2\sqrt{\frac{2GM}{r}}$. Simplifying this equation, we find that $M$ must be increased by a factor of 8 to maintain the condition, hence the mass of the planet would be $8 M_E$.

Question 16

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

Accepted Answer

An astronaut in an orbiting space-craft feels "weightless" because she has the same acceleration as the space-craft. This is due to the fact that both the astronaut and the spacecraft are in free-fall towards the Earth, and therefore experience the same gravitational acceleration. This creates the sensation of weightlessness.

Question 17

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 speed of the object is proportional to the square root of the gravitational force, which depends on the mass of the planet. Since the speed is 4 times greater, the gravitational force must be 16 times greater, implying the mass of the planet is 16 times the mass of Earth.

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 orbit to another, work is done against the gravitational force. The work done is given by the change in potential energy: ΔU = U(final) - U(initial) For a circular orbit, the potential energy is given by: U = -GmM/R where G is the gravitational constant, M is the mass of the Earth, m is the mass of the satellite, and R is the radius of the orbit. When the satellite is moved from an orbit with radius R to an orbit with radius 2R, the potential energy changes by: ΔU = [-GmM/(2R)] - [-GmM/R] ΔU = -GmM/R Therefore, the work done by the gravitational force is equal to the change in potential energy: W = ΔU W = -GmM/R Substituting the given values, we get: W = -G(M)(m)/(2R) W = -GM_Em/2R Hence, the correct answer is E.

Question 19

An object is dropped from an altitude of one Earth radius above Earth's surface.If M is the mass of Earth and R is its radius, the speed of the object just before it hits Earth, neglecting air resistance, is given by:

Accepted Answer

The answer of An object is dropped from an altitude...

Question 20

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...

Quiz 13: Gravitation

A spherical shell has inner radius R₁, outer radius R₂, and mass M, distributed uniformly throughout the shell.The magnitude of the gravitational force exerted on the shell by a point mass m a distance d from the center, inside the inner radius, is:

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?

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?

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 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:

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

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

Mars has a mass of about 0.1075 times the mass of Earth and a diameter of about 0.533 times the diameter of Earth.The acceleration of a body falling near the surface of Mars is about:

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:

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

Each of the four corners of a square with edge a is occupied by a point mass m.There is a fifth mass, also m, at the center of the square.To remove the mass from the center to a point far away the work that must be done by an external agent is given by:

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:

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 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:

An object is dropped from an altitude of one Earth radius above Earth's surface.If M is the mass of Earth and R is its radius, the speed of the object just before it hits Earth, neglecting air resistance, is given by:

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

Quiz 13: Gravitation

A spherical shell has inner radius R1, outer radius R2, and mass M, distributed uniformly throughout the shell.The magnitude of the gravitational force exerted on the shell by a point mass m a distance d from the center, inside the inner radius, is:

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?

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 spherical shell has inner radius R1, outer radius R2, and mass M, distributed uniformly throughout the shell.The magnitude of the gravitational force exerted on the shell by a point particle of mass m, located a distance d from the center, outside the inner radius and inside the outer radius, is:

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 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:

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

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

Mars has a mass of about 0.1075 times the mass of Earth and a diameter of about 0.533 times the diameter of Earth.The acceleration of a body falling near the surface of Mars is about:

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:

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

Each of the four corners of a square with edge a is occupied by a point mass m.There is a fifth mass, also m, at the center of the square.To remove the mass from the center to a point far away the work that must be done by an external agent is given by:

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:

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?

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:

An object is dropped from an altitude of one Earth radius above Earth's surface.If M is the mass of Earth and R is its radius, the speed of the object just before it hits Earth, neglecting air resistance, is given by:

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

A spherical shell has inner radius R₁, outer radius R₂, and mass M, distributed uniformly throughout the shell.The magnitude of the gravitational force exerted on the shell by a point mass m a distance d from the center, inside the inner radius, is:

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: