Chapter 2, Problem 86E

A 2-kg mass is 1 m away from a 5-kg mass. What is the gravitational force (a) that the 5-kg mass exerts on the 2-kg mass, and (b) that the 2-kg mass exerts on the 5-kg mass? (c) If both masses are free to move, what are their respective accelerations if no other forces are acting?

To determine

The gravitational force the $5\text{kg}$ mass exerts on the $2\text{kg}$ mass.

Answer to Problem 86E

The gravitational force the $5\text{kg}$ mass exerts on the $2\text{kg}$ mass is $1.67\times {10}^{-8}\text{N}$.

Explanation of Solution

Given Info: The first mass is $2\text{kg}$, the second mass is $5\text{kg}$ and distance between them is $1\text{m}$.

Write the expression for gravitation force.

$F=\frac{G{m}_1{m}_2}{r^2}$

Here,

$G$ is the gravitational constant.

$m_1$ is the first mass.

$m_2$ is the second mass.

$r$ is the distance between the masses.

Substitute $2\text{kg}$ for $m_1$, $5\text{kg}$ for $m_2$, $1\text{m}$ for $r$ and $6.67\times {10}^{-11}{\text{m}}^3{\text{/kg/s}}^2$ for $G$ to find $F$.

$\begin{array}{c}F=\frac{\left(6.67\times {10}^{-11}{\text{m}}^3{\text{/kg/s}}^2\right)\left(2\text{kg}\right)\left(5\text{kg}\right)}{{\left(1\text{m}\right)}^2}\\ =1.67\times {10}^{-8}\text{N}\end{array}$

Conclusion:

Therefore, the gravitational force the $5\text{kg}$ mass exerts on the $2\text{kg}$ mass is $1.67\times {10}^{-8}\text{N}$.

To determine

The gravitational force the $2\text{kg}$ mass exerts on the $5\text{kg}$ mass.

Answer to Problem 86E

The gravitational force the $2\text{kg}$ mass exerts on the $5\text{kg}$ mass is $1.67\times {10}^{-8}\text{N}$.

Explanation of Solution

Given Info: The first mass is $2\text{kg}$, the second mass is $5\text{kg}$ and distance between them is $1\text{m}$.

Write the expression for gravitation force.

$F=\frac{G{m}_1{m}_2}{r^2}$

Substitute $2\text{kg}$ for $m_1$, $5\text{kg}$ for $m_2$, $1\text{m}$ for $r$ and $6.67\times {10}^{-11}{\text{m}}^3{\text{/kg/s}}^2$ for $G$ to find $F$.

$\begin{array}{c}F=\frac{\left(6.67\times {10}^{-11}{\text{m}}^3{\text{/kg/s}}^2\right)\left(2\text{kg}\right)\left(5\text{kg}\right)}{{\left(1\text{m}\right)}^2}\\ =1.67\times {10}^{-8}\text{N}\end{array}$

Conclusion:

Therefore, the gravitational force the $2\text{kg}$ mass exerts on the $5\text{kg}$ mass is $1.67\times {10}^{-8}\text{N}$.

To determine

The acceleration of the masses.

Answer to Problem 86E

The acceleration of the $2\text{kg}$ mass is $8.3\times {10}^{-9}{\text{m/s}}^2$ and the acceleration of the $5\text{kg}$ mass is $3.3\times {10}^{-9}{\text{m/s}}^2$.

Explanation of Solution

Given Info: The first mass is $2\text{kg}$, the second mass is $5\text{kg}$ and force between them is $1.67\times {10}^{-8}\text{N}$.

Write the expression for acceleration of $2\text{kg}$ mass.

$a_1=\frac{F}{m_1}$

Here,

$F$ is the gravitational force.

$m_1$ is the first mass.

$a_1$ is the acceleration of the first mass.

Substitute $1.67\times {10}^{-8}\text{N}$ for $F$ and $2\text{kg}$ for $m_1$ to find $a_1$.

$\begin{array}{c}{a}_1=\frac{1.67\times {10}^{-8}\text{N}}{2\text{kg}}\\ =8.3\times {10}^{-9}{\text{m/s}}^2\end{array}$

Write the expression for acceleration of $5\text{kg}$ mass.

$a_2=\frac{F}{m_2}$

Here,

$F$ is the gravitational force.

$m_2$ is the second mass.

$a_2$ is the acceleration of the second mass.

Substitute $1.67\times {10}^{-8}\text{N}$ for $F$ and $2\text{kg}$ for $m_1$ to find $a_2$.

$\begin{array}{c}{a}_2=\frac{1.67\times {10}^{-8}\text{N}}{5\text{kg}}\\ =3.3\times {10}^{-9}{\text{m/s}}^2\end{array}$

Conclusion:

Therefore, the acceleration of the $2\text{kg}$ mass is $8.3\times {10}^{-9}{\text{m/s}}^2$ and the acceleration of the $5\text{kg}$ mass is $3.3\times {10}^{-9}{\text{m/s}}^2$.