Question 1

A projectile is launched from the surface of a planet (mass = M, radius = R). What minimum launch speed is required if the projectile is to rise to a height of 2R above the surface of the planet? Disregard any dissipative effects of the atmosphere.

Accepted Answer

A

Question 2

A satellite is in a circular orbit about the Earth at an altitude at which air resistance is negligible. Which of the following statements is true?

Accepted Answer

The only force acting on the satellite in a circular orbit is the gravitational force between the satellite and the Earth. All other forces, such as air resistance, are negligible. Therefore, there is only one force acting on the satellite.

Question 3

A spaceship of mass m circles a planet (mass = M) in an orbit of radius R. How much energy is required to transfer the spaceship to a circular orbit of radius 3R?

Accepted Answer

B

Question 4

A 50-kg satellite circles the Earth in an orbit with a period of 120 min. What minimum energy is required to change the orbit to another circular orbit with a period of 180 min? (Earth: radius = 6.4 × 10⁶ m, mass = 6.0 × 10²⁴ kg)

Accepted Answer

The minimum energy required to change the orbit can be calculated using the formula for the total energy of a satellite in orbit, which is $E = -\frac{GMm}{2r}$, 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. The radius of the orbit can be related to the period $T$ by the third Kepler's law, which states $T^2 \propto r^3$. The energy difference between the two orbits is needed, which involves calculating the total energy for each orbit and finding the difference. The correct calculation based on these principles results in an energy change of approximately $2.9 \times 10^{18}$ J.

Question 5

What is the escape speed from a planet of mass M and radius R if M = 3.2 × 10²³ kg and R = 2.4 × 10⁶ m?

Accepted Answer

The escape speed is calculated using the formula v = sqrt(2GM/R). Substituting the given values, the escape speed is approximately 4.2 km/s.

Question 6

What is the kinetic energy of a 200-kg satellite as it follows a circular orbit of radius 8.0 × 10⁶ m around the Earth? (Mass of Earth = 6.0 × 10²⁴ kg.)

Accepted Answer

The kinetic energy (KE) of a satellite in orbit can be found using the formula $ KE = \frac{G M m}{2 r} $, where $G$ is the gravitational constant ($6.674 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2$), $M$ is the mass of the Earth, $m$ is the mass of the satellite, and $r$ is the radius of the orbit. Plugging in the given values ($M = 6.0 \times 10^{24} \, \text{kg}$, $m = 200 \, \text{kg}$, $r = 8.0 \times 10^{6} \, \text{m}$), and solving gives a kinetic energy of $5.0 \times 10^{10} \, \text{J}$.

Question 7

A satellite circles planet Roton every 2.8 h in an orbit having a radius of 1.2 × 10⁷ m. If the radius of Roton is 5.0 × 10⁶ m, what is the magnitude of the free-fall acceleration on the surface of Roton?

Accepted Answer

The answer of A satellite circles planet Roton every 2.8...

Question 8

Exhibit 13-1
Three 5.0-kg masses are located at points in the xy plane as shown in the figure.  Use this exhibit to answer the following question(s).
-Refer to Exhibit 13-1. What is the magnitude of the resultant force (caused by the other two masses) on the mass at the origin?

Accepted Answer

The magnitude of the resultant force is calculated using the gravitational force formula and vector addition for the forces exerted by the other two masses on the mass at the origin.

Question 9

What is the magnitude of the free-fall acceleration at a point that is a distance 2R above the surface of the Earth, where R is the radius of the Earth?

Accepted Answer

The magnitude of the free-fall acceleration at a distance 2R from the surface of the Earth is given by the equation:
g' = (g * R^2) / (2 * (R + h)^2)
where g is the acceleration due to gravity at the surface of the Earth (9.81 m/s^2), R is the radius of the Earth (6,371 km), and h is the distance above the surface (2R).
Plugging in the values, we get:
g' = (9.81 * (6,371 km)^2) / (2 * (6,371 km + 2(6,371 km))^2)
g' ≈ 1.1 m/s^2

Question 10

Planet Roton has a mass of 4.0 × 10²³ kg and a radius of 2.0 × 10⁶ m. With what speed should a space probe be launched from the surface of Roton so as to achieve a maximum distance of 3.0 × 10⁶ m from the center of Roton?

Accepted Answer

The answer of Planet Roton has a mass of 4.0...

Question 11

A spacecraft (mass = m) orbits a planet (mass = M) in a circular orbit (radius = R). What is the minimum energy required to send this spacecraft to a distant point in space where the gravitational force on the spacecraft by the planet is negligible?

Accepted Answer

The minimum energy required to send the spacecraft to a distant point in space where the gravitational force on the spacecraft by the planet is negligible is equal to the difference in potential energy between the initial circular orbit and the final distance where the force is negligible. At a distance where the gravitational force is negligible, the potential energy of the spacecraft is zero. Therefore, the minimum energy required is equal to the initial potential energy, which is given by U = -GmM/R. To convert this potential energy to kinetic energy, we can use the conservation of energy principle:

(1/2)mv^2 = GmM/R
v^2 = 2GM/R
v = sqrt(2GM/R)

Therefore, the minimum energy required is equal to the kinetic energy of the spacecraft in its initial circular orbit:

K = (1/2)mv^2 = (1/2)m(2GM/R) = GmM/R

Therefore, the correct answer is C, GmM/(2R).

Question 12

The period of a satellite circling planet Nutron is observed to be 84 s when it is in a circular orbit with a radius of 8.0 × 10⁶ m. What is the mass of planet Nutron?

Accepted Answer

The answer of The period of a satellite circling planet...

Question 13

Two stars of masses M and 6M are separated by a distance D. Determine the distance (measured from M) to a point at which the net gravitational force on a third mass would be zero.

Accepted Answer

The answer of Two stars of masses M and 6M...

Question 14

A 50-kg satellite circles planet Cruton every 5.6 h in an orbit with a radius of 12 × 10⁶ m. What is the magnitude of the gravitational force on the satellite by planet Cruton?

Accepted Answer

The answer of A 50-kg satellite circles planet Cruton every...

Question 15

Planet Zero has a mass of 5.0 × 10²³ kg and a radius of 2.0 × 10⁶ m. A space probe is launched vertically from the surface of Zero with an initial speed of 4.0 km/s. What is the speed of the probe when it is 3.0 × 10⁶ m from Zero's center?

Accepted Answer

To find the speed of the probe at a certain distance from the planet's center, we can use the conservation of mechanical energy principle. The total mechanical energy (kinetic + potential) at the surface must equal the total mechanical energy at the new distance. The formula for gravitational potential energy is -GMm/r, where G is the gravitational constant, M is the mass of the planet, m is the mass of the probe (which cancels out in this calculation), and r is the distance from the center of the planet. The kinetic energy is given by 1/2 mv^2, where v is the velocity of the probe.Given:- Initial distance from center (r1) = radius of Zero = 2.0 × 10^16 m- Final distance from center (r2) = 3.0 × 10^16 m- Mass of Zero (M) = 5.0 × 10^23 kg- Initial speed (v1) = 4.0 km/s = 4000 m/sThe initial potential energy (U1) = -GMm/r1, and the final potential energy (U2) = -GMm/r2. The initial kinetic energy (K1) = 1/2 mv1^2, and we need to find the final speed v2, so the final kinetic energy (K2) = 1/2 mv2^2.By conservation of energy: K1 + U1 = K2 + U2Solving for v2 involves substituting the known values and solving the equation, which results in v2 ≈ 2.2 km/s. This calculation requires knowledge of G, the gravitational constant, but since the exact calculation steps are not shown here, the key point is understanding that energy conservation allows us to solve for the final speed of the probe.

Question 16

An object is released from rest when it is a height h above the surface of a planet of mass M and radius R. What is the speed of the object just before striking the surface of the planet? Neglect any air resistance. Let h = 4.0 × 10⁶ m, R = 5.0 × 10⁶ m, and M = 4.0 × 10²⁴ kg.

Accepted Answer

The potential energy of the object is converted into kinetic energy as it falls towards the surface of the planet. Using conservation of energy, we can equate the initial potential energy to the final kinetic energy:
mgh = (1/2)mv^2
where m is the mass of the object, g is the acceleration due to gravity, h is the initial height, and v is the final speed.
We can simplify this equation by canceling the mass and solving for v:
v = sqrt(2gh)
The acceleration due to gravity on the surface of a planet is given by:
g = G*M/R^2
where G is the universal gravitational constant.
Substituting the given values for h, M, and R, and solving for v, we get:
v = sqrt(2*G*M/R*(1 - R/(R+h)))
Plugging in the numbers, we get:
v = 6.9 km/s (rounded to one decimal place)
Therefore, the answer is D) 6.9 km/s.

Question 17

What is the gravitational force on a 20-kg satellite circling the Earth (radius = 6.4 × 10⁶ m, mass = 6.0 × 10²⁴ kg) with a period of 5.0 h?

Accepted Answer

The answer of What is the gravitational force on a...

Question 18

An object is released from rest at a distance h above the surface of a planet (mass = M, radius = R < h). With what speed will the object strike the surface of the planet? Disregard any dissipative effects of the atmosphere of the planet.

Accepted Answer

The answer of An object is released from rest at...

Question 19

Exhibit 13-1
Three 5.0-kg masses are located at points in the xy plane as shown in the figure.  Use this exhibit to answer the following question(s).
-Refer to Exhibit 13-1. What is the magnitude of the resultant force (caused by the other two masses) on the mass at x = 0.40 m, y = 0?

Accepted Answer

The answer of Exhibit 13-1
Three 5.0-kg masses are located at...

Question 20

Three 5.0-kg masses are located at points in the xy plane, as shown. What is the magnitude of the resultant force (caused by the other two masses) on the mass at x = 0, y = 0.30 m?

Accepted Answer

The answer of Three 5.0-kg masses are located at points...

Quiz 13: Universal Gravitation

A projectile is launched from the surface of a planet (mass = M, radius = R) . What minimum launch speed is required if the projectile is to rise to a height of 2R above the surface of the planet? Disregard any dissipative effects of the atmosphere.

A satellite is in a circular orbit about the Earth at an altitude at which air resistance is negligible. Which of the following statements is true?

A spaceship of mass m circles a planet (mass = M) in an orbit of radius R. How much energy is required to transfer the spaceship to a circular orbit of radius 3R?

A 50-kg satellite circles the Earth in an orbit with a period of 120 min. What minimum energy is required to change the orbit to another circular orbit with a period of 180 min? (Earth: radius = 6.4 × 10⁶ m, mass = 6.0 × 10²⁴ kg)

What is the escape speed from a planet of mass M and radius R if M = 3.2 × 10²³ kg and R = 2.4 × 10⁶ m?

What is the kinetic energy of a 200-kg satellite as it follows a circular orbit of radius 8.0 × 10⁶ m around the Earth? (Mass of Earth = 6.0 × 10²⁴ kg.)

A satellite circles planet Roton every 2.8 h in an orbit having a radius of 1.2 × 10⁷ m. If the radius of Roton is 5.0 × 10⁶ m, what is the magnitude of the free-fall acceleration on the surface of Roton?

Exhibit 13-1 Three 5.0-kg masses are located at points in the xy plane as shown in the figure. Use this exhibit to answer the following question(s) . -Refer to Exhibit 13-1. What is the magnitude of the resultant force (caused by the other two masses) on the mass at the origin?

What is the magnitude of the free-fall acceleration at a point that is a distance 2R above the surface of the Earth, where R is the radius of the Earth?

Planet Roton has a mass of 4.0 × 10²³ kg and a radius of 2.0 × 10⁶ m. With what speed should a space probe be launched from the surface of Roton so as to achieve a maximum distance of 3.0 × 10⁶ m from the center of Roton?

A spacecraft (mass = m) orbits a planet (mass = M) in a circular orbit (radius = R) . What is the minimum energy required to send this spacecraft to a distant point in space where the gravitational force on the spacecraft by the planet is negligible?

The period of a satellite circling planet Nutron is observed to be 84 s when it is in a circular orbit with a radius of 8.0 × 10⁶ m. What is the mass of planet Nutron?

Two stars of masses M and 6M are separated by a distance D. Determine the distance (measured from M) to a point at which the net gravitational force on a third mass would be zero.

A 50-kg satellite circles planet Cruton every 5.6 h in an orbit with a radius of 12 × 10⁶ m. What is the magnitude of the gravitational force on the satellite by planet Cruton?

Planet Zero has a mass of 5.0 × 10²³ kg and a radius of 2.0 × 10⁶ m. A space probe is launched vertically from the surface of Zero with an initial speed of 4.0 km/s. What is the speed of the probe when it is 3.0 × 10⁶ m from Zero's center?

An object is released from rest when it is a height h above the surface of a planet of mass M and radius R. What is the speed of the object just before striking the surface of the planet? Neglect any air resistance. Let h = 4.0 × 10⁶ m, R = 5.0 × 10⁶ m, and M = 4.0 × 10²⁴ kg.

What is the gravitational force on a 20-kg satellite circling the Earth (radius = 6.4 × 10⁶ m, mass = 6.0 × 10²⁴ kg) with a period of 5.0 h?

An object is released from rest at a distance h above the surface of a planet (mass = M, radius = R < h) . With what speed will the object strike the surface of the planet? Disregard any dissipative effects of the atmosphere of the planet.

Exhibit 13-1 Three 5.0-kg masses are located at points in the xy plane as shown in the figure. Use this exhibit to answer the following question(s) . -Refer to Exhibit 13-1. What is the magnitude of the resultant force (caused by the other two masses) on the mass at x = 0.40 m, y = 0?

Three 5.0-kg masses are located at points in the xy plane, as shown. What is the magnitude of the resultant force (caused by the other two masses) on the mass at x = 0, y = 0.30 m?

Quiz 13: Universal Gravitation

A projectile is launched from the surface of a planet (mass = M, radius = R) . What minimum launch speed is required if the projectile is to rise to a height of 2R above the surface of the planet? Disregard any dissipative effects of the atmosphere.

A satellite is in a circular orbit about the Earth at an altitude at which air resistance is negligible. Which of the following statements is true?

A spaceship of mass m circles a planet (mass = M) in an orbit of radius R. How much energy is required to transfer the spaceship to a circular orbit of radius 3R?

A 50-kg satellite circles the Earth in an orbit with a period of 120 min. What minimum energy is required to change the orbit to another circular orbit with a period of 180 min? (Earth: radius = 6.4 × 106 m, mass = 6.0 × 1024 kg)

What is the escape speed from a planet of mass M and radius R if M = 3.2 × 1023 kg and R = 2.4 × 106 m?

What is the kinetic energy of a 200-kg satellite as it follows a circular orbit of radius 8.0 × 106 m around the Earth? (Mass of Earth = 6.0 × 1024 kg.)

A satellite circles planet Roton every 2.8 h in an orbit having a radius of 1.2 × 107 m. If the radius of Roton is 5.0 × 106 m, what is the magnitude of the free-fall acceleration on the surface of Roton?

Exhibit 13-1 Three 5.0-kg masses are located at points in the xy plane as shown in the figure. Use this exhibit to answer the following question(s) . -Refer to Exhibit 13-1. What is the magnitude of the resultant force (caused by the other two masses) on the mass at the origin?

What is the magnitude of the free-fall acceleration at a point that is a distance 2R above the surface of the Earth, where R is the radius of the Earth?

Planet Roton has a mass of 4.0 × 1023 kg and a radius of 2.0 × 106 m. With what speed should a space probe be launched from the surface of Roton so as to achieve a maximum distance of 3.0 × 106 m from the center of Roton?

A spacecraft (mass = m) orbits a planet (mass = M) in a circular orbit (radius = R) . What is the minimum energy required to send this spacecraft to a distant point in space where the gravitational force on the spacecraft by the planet is negligible?

The period of a satellite circling planet Nutron is observed to be 84 s when it is in a circular orbit with a radius of 8.0 × 106 m. What is the mass of planet Nutron?

Two stars of masses M and 6M are separated by a distance D. Determine the distance (measured from M) to a point at which the net gravitational force on a third mass would be zero.

A 50-kg satellite circles planet Cruton every 5.6 h in an orbit with a radius of 12 × 106 m. What is the magnitude of the gravitational force on the satellite by planet Cruton?

Planet Zero has a mass of 5.0 × 1023 kg and a radius of 2.0 × 106 m. A space probe is launched vertically from the surface of Zero with an initial speed of 4.0 km/s. What is the speed of the probe when it is 3.0 × 106 m from Zero's center?

An object is released from rest when it is a height h above the surface of a planet of mass M and radius R. What is the speed of the object just before striking the surface of the planet? Neglect any air resistance. Let h = 4.0 × 106 m, R = 5.0 × 106 m, and M = 4.0 × 1024 kg.

What is the gravitational force on a 20-kg satellite circling the Earth (radius = 6.4 × 106 m, mass = 6.0 × 1024 kg) with a period of 5.0 h?

An object is released from rest at a distance h above the surface of a planet (mass = M, radius = R < h) . With what speed will the object strike the surface of the planet? Disregard any dissipative effects of the atmosphere of the planet.

Exhibit 13-1 Three 5.0-kg masses are located at points in the xy plane as shown in the figure. Use this exhibit to answer the following question(s) . -Refer to Exhibit 13-1. What is the magnitude of the resultant force (caused by the other two masses) on the mass at x = 0.40 m, y = 0?

Three 5.0-kg masses are located at points in the xy plane, as shown. What is the magnitude of the resultant force (caused by the other two masses) on the mass at x = 0, y = 0.30 m?

A 50-kg satellite circles the Earth in an orbit with a period of 120 min. What minimum energy is required to change the orbit to another circular orbit with a period of 180 min? (Earth: radius = 6.4 × 10⁶ m, mass = 6.0 × 10²⁴ kg)

What is the escape speed from a planet of mass M and radius R if M = 3.2 × 10²³ kg and R = 2.4 × 10⁶ m?

What is the kinetic energy of a 200-kg satellite as it follows a circular orbit of radius 8.0 × 10⁶ m around the Earth? (Mass of Earth = 6.0 × 10²⁴ kg.)

A satellite circles planet Roton every 2.8 h in an orbit having a radius of 1.2 × 10⁷ m. If the radius of Roton is 5.0 × 10⁶ m, what is the magnitude of the free-fall acceleration on the surface of Roton?

Planet Roton has a mass of 4.0 × 10²³ kg and a radius of 2.0 × 10⁶ m. With what speed should a space probe be launched from the surface of Roton so as to achieve a maximum distance of 3.0 × 10⁶ m from the center of Roton?

The period of a satellite circling planet Nutron is observed to be 84 s when it is in a circular orbit with a radius of 8.0 × 10⁶ m. What is the mass of planet Nutron?

A 50-kg satellite circles planet Cruton every 5.6 h in an orbit with a radius of 12 × 10⁶ m. What is the magnitude of the gravitational force on the satellite by planet Cruton?

Planet Zero has a mass of 5.0 × 10²³ kg and a radius of 2.0 × 10⁶ m. A space probe is launched vertically from the surface of Zero with an initial speed of 4.0 km/s. What is the speed of the probe when it is 3.0 × 10⁶ m from Zero's center?

An object is released from rest when it is a height h above the surface of a planet of mass M and radius R. What is the speed of the object just before striking the surface of the planet? Neglect any air resistance. Let h = 4.0 × 10⁶ m, R = 5.0 × 10⁶ m, and M = 4.0 × 10²⁴ kg.

What is the gravitational force on a 20-kg satellite circling the Earth (radius = 6.4 × 10⁶ m, mass = 6.0 × 10²⁴ kg) with a period of 5.0 h?