Question 1

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 2

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 3

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 4

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

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Question 5

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 does not depend on the mass of the projectile, only on the mass of the planet it is escaping from and the distance from the planet's center. Therefore, the escape velocity for the 2.0 kg projectile is also 11 km/s, as it is escaping from the same planet (Earth) at the same distance.

Question 6

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

Accepted Answer

At the center of the Earth, an object experiences weightlessness due to the gravitational pull of the mass surrounding the object being equal in all directions.

Question 7

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 8

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

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Question 9

For a planet in orbit around a star the perihelion distance is r_p and its speed at perihelion is v_p.The aphelion distance is r_a and its speed at aphelion is v_a.Which of following is true?

Accepted Answer

The correct answer is based on the conservation of angular momentum in an elliptical orbit, where the product of the speed and the radial distance at any two points (such as perihelion and aphelion) remains constant. This is expressed as $v_{\text{a}} r_{\text{a}} = v_{\text{p}} r_{\text{p}}$, where $v_{\text{a}}$ and $v_{\text{p}}$ are the speeds at aphelion and perihelion, respectively, and $r_{\text{a}}$ and $r_{\text{p}}$ are the corresponding radial distances.

Question 10

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

During the move, the gravitational force is in the opposite direction to the displacement of the satellite (from R to 2R), therefore the angle between the force and displacement vectors is 180 degrees. In this case, the work done by the gravitational force is negative.

Question 11

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

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Question 12

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 the external agent is equal to the gravitational potential energy of the system before and after bringing the third particle to the resting point. Before the third particle is brought to the resting point, the potential energy of the system is:

U1 = -Gm(2m)/d - Gm(2m)/d = -4Gm^2/d

After the third particle is brought to the resting point, the potential energy of the system is:

U2 = -Gm(2m)/d + Gm(2m)/2d + Gm(2m)/2d = -8Gm^2/d

Therefore, the work done by the external agent is:

W = U2 - U1 = (-8Gm^2/d) - (-4Gm^2/d) = -4Gm^2/d

Since the work done is negative, choice D is the correct answer.

Question 13

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

Accepted Answer

The astronaut is in orbit around the Earth, which means that both she and the spacecraft are falling towards the Earth due to the influence of gravity. However, because the spacecraft is moving forward at a high enough speed, it's constantly "missing" the Earth and falling "over" it, resulting in a circular or elliptical orbit. This means that the astronaut is in a state of freefall, just like objects falling towards the Earth's surface. As a result, she experiences no sensation of weight or heaviness, and instead feels weightless. This is because she has the same acceleration as the spacecraft, and so no forces are acting on her that would cause her to feel any weight or resistance.

Question 14

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 15

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 16

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 17

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

Accepted Answer

According to Kepler's laws of planetary motion, a comet moves faster as it approaches the sun and slower as it recedes. Therefore, the speed of a comet in an elliptical orbit about the sun decreases while it is receding from the sun.

Question 18

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 19

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 20

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.

Quiz 13: Gravitation

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:

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:

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?

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:

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

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:

For a planet in orbit around a star the perihelion distance is r_p and its speed at perihelion is v_p.The aphelion distance is r_a and its speed at aphelion is v_a.Which of following is true?

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?

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:

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

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, 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,

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

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

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

Quiz 13: Gravitation

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:

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:

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?

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:

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

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:

For a planet in orbit around a star the perihelion distance is rp and its speed at perihelion is vp.The aphelion distance is ra and its speed at aphelion is va.Which of following is true?

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?

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:

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

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, 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,

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

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

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

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:

For a planet in orbit around a star the perihelion distance is r_p and its speed at perihelion is v_p.The aphelion distance is r_a and its speed at aphelion is v_a.Which of following is true?

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?