Question 1

The constant of universal gravitation is A) ~9.81 m/s² B) ~6.67 $\times$ 10^-11 N · m²/kg² C) ~the gravitational attraction between the masses of any two bodies. D) ~the ratio of the forces of attraction between any two masses. E) None of these is correct.

Accepted Answer

The constant of universal gravitation is defined as the force of attraction between two point masses of 1 kilogram each that are separated by a distance of 1 meter. Its value is approximately 6.67 x 10^-11 N · m^2/kg^2. This means that the gravitational force between two point masses (of any sizes) can be calculated using this constant, their masses, and the distance between them. Option A is the acceleration due to gravity on the Earth's surface. Options C and D are not accurate definitions of the constant of universal gravitation.

Question 2

A certain object weighs 22.2 N on the surface of Earth. If the radius of the moon is 0.276 times the radius of Earth and the mass of the moon is 0.0123 times the mass of Earth, the object's weight on the surface of the moon is approximately
A) 0.365 N
B) 3.58 N
C) 9.79 N
D) 22.2 N
E) 133 N

Accepted Answer

Weight=mass×acceleration due to gravity
On Earth,
Weight=22.2 N
Mass of object=m
Acceleration due to gravity on the Earth=g
So, 22.2=m×g             ---(1)
On the moon,
Mass of object=m
Acceleration due to gravity on the moon=g/6       (as gmoon=1/6gEarth)
Radius of moon=0.276×Radius of Earth
Mass of moon=0.0123×Mass of Earth
Weight of an object on the moon=w
So, w=m×g/6                      ---(2)
Dividing equation 1 by equation 2,
(22.2/w)=6(g/gmoon)
Putting the values of g/gmoon, we get
(22.2/w)=6(1/6)
w=3.58 N

Therefore, the weight of the object on the lunar surface is approximately 3.58 N.

Question 3

A planet is made of two distinct materials. From the core to R/2, the density of the material is 4000 kg/m³, and from R/2 to R the density is 3000 kg/m³. What is the gravity at the surface of the planet if R = 5000 km? A) 0.699 m/s² B) 3.66 m/s² C) 4.19 m/s² D) 4.38 m/s² E) 4.99 m/s²

Accepted Answer

The answer of A planet is made of two distinct...

Question 4

Kepler's second law, "a line joining any planet to the Sun sweeps out equal areas in equal times" is based on which conservation principle?
A) conservation of energy
B) conservation of linear momentum
C) conservation of angular momentum
D) the work-energy theorem
E) It is not based on any conservation principle.

Accepted Answer

Kepler's second law is based on the conservation of angular momentum. This means that as a planet moves closer to the Sun, it speeds up and as it moves farther away it slows down to ensure that the product of its velocity and its distance from the Sun stays constant. This results in equal areas being swept out in equal amounts of time.

Question 5

Suppose the gravitational mass, m_G, and inertial mass, m_I, were not equivalent and 1m_G = 2 m_I. Near Earth's surface, the acceleration due to gravity would have a value of A) 4.905 m/s² B) 9.81 m/s² C) 14.7 m/s² D) 19.62 m/s² E) 24.53 m/s²

Accepted Answer

The acceleration due to gravity is calculated using the formula $a = \frac{F}{m_{I}}$, where $F = G \frac{m_{E}m_{G}}{r^2}$ and $m_{I}$ is the inertial mass. If $m_{G} = 2m_{I}$, then $a = G \frac{m_{E}2m_{I}}{r^2} \cdot \frac{1}{m_{I}} = 2g$, where $g = 9.81 m/s^2$ is the standard acceleration due to gravity. Thus, $a = 2 \times 9.81 m/s^2 = 19.62 m/s^2$. However, since the question implies a direct relationship between gravitational and inertial mass that contradicts this setup, and no option matches the calculated value, the premise of the question suggests a misunderstanding. Given the standard understanding that gravitational and inertial masses are equivalent, and without altering the fundamental laws of physics, the acceleration due to gravity remains $9.81 m/s^2$, making all provided options incorrect based on the calculation. However, the question's premise itself is flawed as it contradicts the equivalence principle, which is a cornerstone of general relativity, stating that gravitational and inertial masses are equivalent. This principle ensures that the acceleration due to gravity is constant and equal to $9.81 m/s^2$ near Earth's surface, not altered by hypothetical differences between gravitational and inertial mass.

Question 6

When a falling meteor is at a distance of twice Earth's radius (r_e = 6.436 $\times$ 10⁶ m) from the surface of Earth, its acceleration due to gravity is approximately A) 9.80 m/s² B) 4.90 m/s² C) 3.27 m/s² D) 2.45 m/s² E) 1.09 m/s²

Accepted Answer

The answer of When a falling meteor is at a...

Question 7

According to Newton's law of universal gravitation, if the distance between two bodies is tripled, the gravitational force between them is
A) unchanged.
B) halved.
C) doubled.
D) reduced to 1/3 its previous value.
E) None of these is correct.

Accepted Answer

Newton's law of universal gravitation states that the gravitational force between two bodies is inversely proportional to the square of the distance between them. Therefore, if the distance is tripled, the force is reduced to 1/9 of its original value, which is not an option provided.

Question 8

The Cavendish experiment has been described in elementary textbooks as "weighing the Earth." In fact, it was designed to measure the
A) acceleration due to gravity.
B) mass of Earth.
C) period of an Earth satellite.
D) gravitational constant G.
E) variation of g with height.

Accepted Answer

The Cavendish experiment was designed to measure the gravitational constant G, which relates the force of gravity to the masses and distance between them. It did not directly measure the mass of Earth or the acceleration due to gravity, although these values could be calculated using known values of G and the distance between Earth and the apparatus used in the experiment. The experiment also did not involve the period of an Earth satellite or the variation of gravity with height.

Question 9

At the surface of the moon, the acceleration due to the gravity of the moon is  . At a distance from the center of the moon equal to three times the radius of the moon, the acceleration due to the gravity of the moon is
A) 9 
B)  /3
C)  /4
D)  /9
E) 27

Accepted Answer

According to the inverse square law of gravity, the acceleration due to gravity is inversely proportional to square of distance from the center of the celestial body. Therefore, the acceleration due to gravity at a distance equal to three times the radius of the moon is:
Acceleration =  (1/3)^2 × Acceleration at the surface of the moon
Acceleration = 1/9 × 11eaa4be_84f8_c9a2_9e4a_59140f2a7464_TB6080_11
Acceleration = 11eaa4be_84f9_3ed6_9e4a_dfdc932c4896_TB6080_11 /9
Hence, the correct choice is D.

Question 10

Newton's law of gravity is F = GMm/r². What are the SI units of the gravitational constant G? A) G has no dimensions B) N · kg²/m² C) N · m²/kg² D) kg²/N · m² E) None of these is correct.

Accepted Answer

In the equation F = GMm/r^2, force (F) is measured in newtons (N), mass (m) is measured in kilograms (kg), and radius (r) is measured in meters (m). Rearranging the equation to solve for G, we get G = Fr^2/Mm. Plugging in the units we get: G = (N * m^2/kg^2). Therefore, the SI units of G are N·m^2/kg^2, which is equivalent to N·m/kg^2. The correct answer is C, N·m/kg^2.

Question 11

In the SI system, the units for gravitational field are A) kg · m/s² B) m/s² C) kg²/m² D) N/m E) None of these is correct.

Accepted Answer

The answer of In the SI system, the units for...

Question 12

You need an expression for the acceleration of the moon toward Earth. If the mass of the earth is M_e, the mass of the moon M_m, the separation of Earth and moon r, and the appropriate gravitational constant is G, the correct expression for the moon's acceleration is A) GM_eM_m/r² B) GM_eM_m²/r² C) GM_m/r² D) GM_e/r² E) GM_e/r²M_m

Accepted Answer

The acceleration of the moon toward Earth is given by the formula:

a = GMm/r^2

where a is the acceleration, G is the gravitational constant, M is the mass of the Earth, m is the mass of the moon, and r is the separation between the centers of the Earth and the moon.

Option D correctly represents this formula, with GM/r^2 representing the gravitational force between the two bodies and Mm/r^2 being the gravitational acceleration experienced by the moon due to this force.

Therefore, choice D is the best answer.

Question 13

A planet is made of two distinct materials in two layers. From the core to R/2, the density of the material is 4000 kg/m³, and from R/2 to R the density is 3000 kg/m³. What is the mass of the planet if R = 5000 km? A) 2.62 $\times$ 10²³ kg B) 1.37 $\times$ 10²⁴ kg C) 1.64 $\times$ 10²⁴ kg D) 1.57 $\times$ 10²⁴ kg E) 1.87 $\times$ 10²⁴ kg

Accepted Answer

We can use the formula for the mass of a planet with two distinct materials in two layers:

$M = \frac{4}{3}\pi R^3 (ho_1 \frac{R}{2}+ho_2 \frac{R}{2})$

Plugging in the given values, we get:

$M = \frac{4}{3}\pi (5000 	ext{ km})^3 (4000 	ext{ kg/m}^3 \frac{5000	ext{ km}}{2} + 3000 	ext{ kg/m}^3 \frac{5000	ext{ km}}{2})$

Simplifying and converting units, we get:

$M = 1.64 	imes 10^{24} 	ext{ kg}$

Therefore, the answer is (C).

Question 14

When two masses are a distance R apart, each exerts a force of magnitude F on the other. When the distance between them is changed to 4R, the force is changed to
A) 16F
B) 4F
C) F/2
D) F/4
E) F/16

Accepted Answer

The force between two masses is given by Newton's law of universal gravitation, which states that the force is inversely proportional to the square of the distance between the two masses. Therefore, if the distance is increased to 4R, the force becomes $F/(4^2) = F/16$.

Question 15

If the mass of Earth is 6 $\times$ 10²⁴ kg, the mass of the moon 7 $\times$ 10²² kg, the radius of the moon's orbit 4 $\times$10⁸ m, and the value of the gravitational constant 6 $\times$ 10^-11 N · m²/kg², the force between Earth and the moon is approximately A) 5 $\times$ 10⁴ N B) 2 $\times$ 10²⁰ N C) 3 $\times$ 10⁵⁰ N D) 7 $\times$ 10³⁰ N E) 3 $\times$ 10²⁸ N

Accepted Answer

The answer of If the mass of Earth is 6...

Question 16

If a planet has a mass twice that of Earth and a radius four times that of Earth, the ratio of the acceleration due to gravity on the planet to that on Earth is
A) 1/8
B) 1/2
C) 1/16
D) 2/1
E) 12/1

Accepted Answer

The answer of If a planet has a mass twice...

Question 17

The acceleration due to gravity in the vicinity of Earth
A) varies directly with the distance from the center of Earth.
B) is a constant that is independent of altitude.
C) varies inversely with the distance from the center of Earth.
D) varies inversely with the square of the distance from the center of Earth.
E) is described by none of these.

Accepted Answer

The acceleration due to gravity is inversely proportional to the square of the distance from the center of the Earth, according to the law of universal gravitation. This means that as one gets farther from the center of the Earth, the acceleration due to gravity decreases at a rate proportional to the square of the distance.

Question 18

Suppose a planet exists that has half the mass of Earth and half its radius. On the surface of that planet, the acceleration due to gravity is
A) twice that on Earth.
B) the same as that on Earth.
C) half that on Earth.
D) one-fourth that on Earth.
E) none of these.

Accepted Answer

The acceleration due to gravity on the surface of a planet is given by the formula g = G (M/R^2), where G is the universal gravitational constant, M is the mass of the planet, and R is its radius. Since this planet has half the mass and half the radius of Earth, its acceleration due to gravity would be g = G ((1/2)M/(1/2R)^2) = G (M/R^2) x 2. This means that the acceleration due to gravity on this planet would be twice that on Earth.

Question 19

The mean radius of Earth is about 6.436 $\times$ 10⁶ m. A satellite in a circular orbit 3.20 $\times$ 10⁶ m above the surface of Earth has an acceleration of approximately A) 39.2 m/s² B) 22.1 m/s² C) 14.2 m/s² D) 6.54 m/s² E) 4.36 m/s²

Accepted Answer

The answer of The mean radius of Earth is about...

Question 20

The acceleration due to gravity at the surface of Earth is g. The radius of Earth is R_E. The distance from the center of Earth to a point where the acceleration due to gravity is g/9 is A) R_E B) 9R_E C) R_E/3 D) 3R_E E) None of these is correct.

Accepted Answer

The acceleration due to gravity at a distance $r$ from the center of a planet of radius $R$ and mass $M$ is given by $g' = \frac{G M}{r^2}$, where $G$ is the gravitational constant. At the surface of the Earth, this acceleration is $g = \frac{G M}{R^2}$. To find the distance $r$ where the acceleration due to gravity is $g/9$, we set $g' = g/9$ and solve for $r$, leading to $r = 3R$, where $R$ is the radius of the Earth.

Quiz 11: Gravity

The constant of universal gravitation is

A certain object weighs 22.2 N on the surface of Earth. If the radius of the moon is 0.276 times the radius of Earth and the mass of the moon is 0.0123 times the mass of Earth, the object's weight on the surface of the moon is approximately

A planet is made of two distinct materials. From the core to R/2, the density of the material is 4000 kg/m³, and from R/2 to R the density is 3000 kg/m³. What is the gravity at the surface of the planet if R = 5000 km?

Kepler's second law, "a line joining any planet to the Sun sweeps out equal areas in equal times" is based on which conservation principle?

Suppose the gravitational mass, m_G, and inertial mass, m_I, were not equivalent and 1m_G = 2 m_I. Near Earth's surface, the acceleration due to gravity would have a value of

When a falling meteor is at a distance of twice Earth's radius (r_e = 6.436 $\times$ 10⁶ m) from the surface of Earth, its acceleration due to gravity is approximately

According to Newton's law of universal gravitation, if the distance between two bodies is tripled, the gravitational force between them is

The Cavendish experiment has been described in elementary textbooks as "weighing the Earth." In fact, it was designed to measure the

At the surface of the moon, the acceleration due to the gravity of the moon is . At a distance from the center of the moon equal to three times the radius of the moon, the acceleration due to the gravity of the moon is

Newton's law of gravity is F = GMm/r². What are the SI units of the gravitational constant G?

In the SI system, the units for gravitational field are

You need an expression for the acceleration of the moon toward Earth. If the mass of the earth is M_e, the mass of the moon M_m, the separation of Earth and moon r, and the appropriate gravitational constant is G, the correct expression for the moon's acceleration is

A planet is made of two distinct materials in two layers. From the core to R/2, the density of the material is 4000 kg/m³, and from R/2 to R the density is 3000 kg/m³. What is the mass of the planet if R = 5000 km?

When two masses are a distance R apart, each exerts a force of magnitude F on the other. When the distance between them is changed to 4R, the force is changed to

If the mass of Earth is 6 $\times$ 10²⁴ kg, the mass of the moon 7 $\times$ 10²² kg, the radius of the moon's orbit 4 $\times$ 10⁸ m, and the value of the gravitational constant 6 $\times$ 10^-11 N · m²/kg², the force between Earth and the moon is approximately

If a planet has a mass twice that of Earth and a radius four times that of Earth, the ratio of the acceleration due to gravity on the planet to that on Earth is

The acceleration due to gravity in the vicinity of Earth

Suppose a planet exists that has half the mass of Earth and half its radius. On the surface of that planet, the acceleration due to gravity is

The mean radius of Earth is about 6.436 $\times$ 10⁶ m. A satellite in a circular orbit 3.20 $\times$ 10⁶ m above the surface of Earth has an acceleration of approximately

The acceleration due to gravity at the surface of Earth is g. The radius of Earth is R_E. The distance from the center of Earth to a point where the acceleration due to gravity is g/9 is