What is the distance from the center of the Moon to the point between Earth and the Moon where the gravitational pulls of Earth and Moon are equal? The mass of Earth is $5.97 \times 10 ^ { 24 } \mathrm {~kg}$ the mass of the Moon is $7.35 \times 10 ^ { 22 } \mathrm {~kg}$ the center-to-center distance between Earth and the Moon is 3.84 \times $10 ^ { 8 } \mathrm {~m}$ and $G = 6.67 \times 10 ^ { - 11 } \mathrm {~N} \cdot \mathrm { m } ^ { 2 } / \mathrm { kg } ^ { 2 }$
A) $3.84 \times 10 ^ { 7 } \mathrm {~m}$
B) $3.83 \times 10 ^ { 6 } \mathrm {~m}$
C) $4.69 \times 10 ^ { 7 } \mathrm {~m}$
D) $4.69 \times 10 ^ { 6 } \mathrm {~m}$
E) $3.45 \times 10 ^ { 8 } \mathrm {~m}$

Question

What is the distance from the center of the Moon to the point between Earth and the Moon where the gravitational pulls of Earth and Moon are equal? The mass of Earth is $5.97 \times 10 ^ { 24 } \mathrm {~kg}$ the mass of the Moon is $7.35 \times 10 ^ { 22 } \mathrm {~kg}$ the center-to-center distance between Earth and the Moon is  3.84 \times  $10 ^ { 8 } \mathrm {~m}$ and $G = 6.67 \times 10 ^ { - 11 } \mathrm {~N} \cdot \mathrm { m } ^ { 2 } / \mathrm { kg } ^ { 2 }$
A) $3.84 \times 10 ^ { 7 } \mathrm {~m}$
B) $3.83 \times 10 ^ { 6 } \mathrm {~m}$
C) $4.69 \times 10 ^ { 7 } \mathrm {~m}$
D) $4.69 \times 10 ^ { 6 } \mathrm {~m}$
E) $3.45 \times 10 ^ { 8 } \mathrm {~m}$

Quizplus · Accepted Answer

The gravitational force between two objects is given by the equation:$$F = G \frac{m_1 m_2}{r^2}$$where:* $F$ is the gravitational force in newtons* $G$ is the gravitational constant, which is $6.67 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^2 / \mathrm{kg}^2$* $m_1$ and $m_2$ are the masses of the two objects in kilograms* $r$ is the distance between the centers of the two objects in metersIn this problem, we are given the masses of the Earth and the Moon, as well as the distance between their centers. We want to find the distance from the center of the Moon to the point between Earth and the Moon where the gravitational pulls of Earth and Moon are equal.Let's call this distance $x$. The gravitational force between the Earth and the Moon is:$$F_E = G \frac{m_E m_M}{(x + 3.84 \times 10^8 \mathrm{~m})^2}$$where $m_E$ is the mass of the Earth and $m_M$ is the mass of the Moon.The gravitational force between the Moon and the point $x$ is:$$F_M = G \frac{m_M m_x}{x^2}$$where $m_x$ is the mass of the point $x$.Since the gravitational pulls of Earth and Moon are equal at point $x$, we have:$$F_E = F_M$$Substituting the expressions for $F_E$ and $F_M$, we get:$$G \frac{m_E m_M}{(x + 3.84 \times 10^8 \mathrm{~m})^2} = G \frac{m_M m_x}{x^2}$$Canceling out the common factors, we get:$$\frac{m_E}{(x + 3.84 \times 10^8 \mathrm{~m})^2} = \frac{m_x}{x^2}$$Solving for $x$, we get:$$x = \frac{m_E}{m_x} \cdot (3.84 \times 10^8 \mathrm{~m})^2$$Substituting the given values for $m_E$ and $m_x$, we get:$$x = \frac{5.97 \times 10^{24} \mathrm{~kg}}{7.35 \times 10^{22} \mathrm{~kg}} \cdot (3.84 \times 10^8 \mathrm{~m})^2 = 3.84 \times 10^7 \mathrm{~m}$$Therefore, the distance from the center of the Moon to the point between Earth and the Moon where the gravitational pulls of Earth and Moon are equal is $3.84 \times 10^7 \mathrm{~m}$.

What Is the Distance from the Center of the Moon 5.97×1024 kg5.97 \times 10 ^ { 24 } \mathrm {~kg}5.97×1024 kg

What Is the Distance from the Center of the Moon $5.97 \times 10 ^ { 24 } \mathrm {~kg}$