Question 1

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? A)4.8 m/s² B)1.1 m/s² C)3.3 m/s² D)2.5 m/s² E)6.5 m/s²

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 2

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.
A)7.8 km/s
B)3.5 km/s
C)5.4 km/s
D)6.9 km/s
E)4.8 km/s

Accepted Answer

The potential energy of the object at a height h above the surface of the planet can be calculated as:
PE = GMm/r
where G is the gravitational constant, M is the mass of the planet, m is the mass of the object, and r = R + h is the distance between the center of the planet and the object.
The kinetic energy of the object just before striking the surface of the planet is given by:
KE = 1/2 mv^2
where v is the velocity of the object.
According to the law of conservation of energy, the total mechanical energy of the system (PE + KE) must be conserved. Therefore,
GMm/r = 1/2 mv^2
Simplifying and solving for v, we get:
v = sqrt(2GM/r)
Substituting the given values, we get:
v = sqrt(2*6.67*10^-11*4*10^24/(5*10^6+4*10^6))
v ≈ 6.89 km/s
Therefore, the speed of the object just before striking the surface of the planet is approximately 6.9 km/s. The correct answer is D.

Question 3

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?
A)63 N
B)58 N
C)68 N
D)73 N
E)50 N

Accepted Answer

B

Question 4

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).
-What is the magnitude of the resultant force (caused by the other two masses) on the mass at x = 0.40 m, y = 0?
A)2.2 × 10−8 N
B)1.9 × 10−8 N
C)1.4 × 10−8 N
D)1.6 × 10−8 N
E)2.5 × 10−8 N

Accepted Answer

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

Question 5

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).
-What is the magnitude of the resultant force (caused by the other two masses) on the mass at the origin?
A)2.7 × 10−8 N
B)2.1 × 10−8 N
C)1.8 × 10−8 N
D)2.4 × 10−8 N
E)2.9 × 10−8 N

Accepted Answer

Using the law of gravitation, we can calculate the force due to each mass on the mass at the origin. Then we can use vector addition to find the resultant force. The force due to the mass at (0,5) is 2.7 x 10^-8 N, the force due to the mass at (5,0) is also 2.7 x 10^-8 N (since the distance is the same), and the force due to the mass at (0,-5) is 2.1 x 10^-8 N. Using vector addition, we can find the magnitude of the resultant force to be approximately 2.1 x 10^-8 N. Therefore, the best choice is B.

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.)
A)5.0 × 109 J
B)1.0 × 1010 J
C)1.5 × 1010 J
D)2.0 × 1010 J
E)2.5 × 109 J

Accepted Answer

The kinetic energy (KE) of a satellite in circular orbit is given by KE = (1/2) * (G * M * m) / r, where G is the gravitational constant (6.674 × 10^-11 N(m/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 values, KE = (1/2) * (6.674 × 10^-11) * (6.0 × 10^24) * (200) / (8.0 × 10^6) = 5.0 × 10^9 J.

Question 7

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?
A)5.5 km/s
B)4.2 km/s
C)5.2 km/s
D)4.8 km/s
E)3.7 km/s

Accepted Answer

The escape speed $ v_e $ is calculated using the formula $ v_e = \sqrt{\frac{2GM}{r}} $. Substituting the given values $ G = 6.674 \times 10^{-11} \, \text{m}^3/\text{kg}\cdot\text{s}^2 $, $ M = 3.2 \times 10^{23} \, \text{kg} $, and $ r = 2.4 \times 10^6 \, \text{m} $, the escape speed is approximately 4.2 km/s.

Question 8

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?
A)3.0 km/s
B)2.2 km/s
C)1.6 km/s
D)3.7 km/s
E)5.9 km/s

Accepted Answer

Firstly, we need to apply the conservation of energy principle, which states that the sum of kinetic and potential energy of a system is always constant. At the surface of Zero, the probe has kinetic energy (KE) given by:

KE = (1/2)mv^2 = (1/2)(m)(4.0 × 10^3 m/s)^2
KE = 8.0 × 10^10 m^2/s^2

At a distance of 3.0 × 10^6 m from Zero's center, the potential energy (PE) of the probe becomes:

PE = G(Mm)/r
PE = (6.67 × 10^-11 N(m/kg)^2)(m)(M)/r
PE = (6.67 × 10^-11 N(m/kg)^2)(m)((5.0 × 10^23 kg))/((3.0 × 10^6 m))
PE = 5.6 × 10^10 m^2/s^2

Where G is the gravitational constant, M is the mass of Zero, m is the mass of the probe, and r is the distance between their centers.

The sum of kinetic and potential energy remains constant, thus:

KE + PE = constant

At the surface, the sum is:

8.0 × 10^10 m^2/s^2 + 0 = 8.0 × 10^10 m^2/s^2

At a distance of 3.0 × 10^6 m, the sum is:

KE + PE = 8.0 × 10^10 m^2/s^2 + 5.6 × 10^10 m^2/s^2
KE + PE = 1.4 × 10^11 m^2/s^2

We can then solve for the speed of the probe v:

v = sqrt((2/m)(KE + PE))

Where m is the mass of the probe.

v = sqrt((2/m)(1.4 × 10^11 m^2/s^2))
v = sqrt((2/(m))(1.4 × 10^11 m^2/s^2))
v = sqrt((2/(m))(1.4 × 10^11 m^2/s^2))
v = 2.2 × 10^3 m/s = 2.2 km/s

Therefore, the speed of the probe when it is 3.0 × 10^6 m from Zero's center is 2.2 km/s.

Question 9

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? A)31 m/s² B)27 m/s² C)34 m/s² D)40 m/s² E)19 m/s²

Accepted Answer

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

Question 10

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.
A) 
B)​ 
C) 
D) 
E)

Accepted Answer

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

Question 11

A spaceship of mass m1 circles a planet (mass = m²) in an orbit of radius r. How much energy is required to transfer the spaceship to a circular orbit of radius 3R? A)Gm1m²/(2r) B) Gm1m²/(3r) C) Gm1m²/(4r) D) Gm1m²/(6r) E)3Gm1m²/(4r)

Accepted Answer

The answer of A spaceship of mass m1 circles a...

Question 12

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)
A)2.9 × 108 J
B)3.5 × 108 J
C)4.1 × 108 J
D)4.7 × 108 J
E)5.9 × 108 J

Accepted Answer

The energy required to change the orbit can be calculated using the formula for gravitational potential energy and the vis-viva equation. The energy difference between the two orbits is found by calculating the total energy for each orbit and subtracting them. The correct calculation gives approximately 2.9 × 10^8 J.

Question 13

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)4.2 km/s
B)3.9 km/s
C)3.0 km/s
D)3.4 km/s
E)6.0 km/s

Accepted Answer

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

Question 14

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)2.6 × 10−8 N
B)2.0 × 10−8 N
C)2.9 × 10−8 N
D)2.3 × 10−8 N
E)2.1 × 10−8 N

Accepted Answer

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

Question 15

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) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of A projectile is launched from the surface...

Question 16

A spacecraft (mass = m₁) orbits a planet (mass = m₂) 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? A)Gm₁m₂/(4r) B)Gm₁m₂/R C)Gm₁m₂/(2R) D)Gm₁m₂/(3R) E)2Gm₁m₂/(5R)

Accepted Answer

The minimum energy required to send the spacecraft to a distant point in space where the gravitational force on it by the planet is negligible is equal to the energy required to overcome the planet's gravitational attraction and move the spacecraft from its initial circular orbit to an infinite distance away from the planet. This is equal to the difference in potential energy of the spacecraft between these two points, which is equal to the initial potential energy minus the final potential energy. The initial potential energy of the spacecraft in its circular orbit is equal to -Gm₁m₂/r, and the final potential energy when the spacecraft is at an infinite distance from the planet is equal to zero. Therefore, the minimum energy required is equal to the initial potential energy, which simplifies to Gm₁m₂/(2R). The correct answer is therefore C.

Question 17

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)6.2 × 1028 kg
B)5.0 × 1028 kg
C)5.5 × 1028 kg
D)4.3 × 1028 kg
E)3.7 × 1028 kg

Accepted Answer

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

Question 18

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)0.41 D
B)0.33 D
C)0.37 D
D)0.29 D
E)0.14 D

Accepted Answer

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

Question 19

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?
A)88 N
B)55 N
C)36 N
D)98 N
E)18 N

Accepted Answer

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

Question 20

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)There is only one force acting on the satellite.
B)There are two forces acting on the satellite, and their resultant is zero.
C)There are two forces acting on the satellite, and their resultant is not zero.
D)There are three forces acting on the satellite.
E)None of the preceding statements are correct.

Accepted Answer

The only force acting on the satellite in a circular orbit is the gravitational force between the satellite and the Earth. Since air resistance is negligible, there are no other forces acting on the satellite.

Quiz 7: Gravity

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?

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.

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?

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) . -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 in the figure. Use this exhibit to answer the following question(s) . -What is the magnitude of the resultant force (caused by the other two masses) on the mass at the origin?

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.)

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?

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?

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?

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.

A spaceship of mass m1 circles a planet (mass = m²) 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)

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?

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 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 spacecraft (mass = m₁) orbits a planet (mass = m₂) 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.

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?

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?