Chapter 34, Problem 74PQ

To determine

The three type of forces acting on the sail.

Answer to Problem 74PQ

The gravitational force acting on the sail is $30\text{N}$. The drag force acting on the sail is $1.4\times {10}^{-5}\text{N}$. The force exerted by solar radiations on the sail is $9\times {10}^{-5}\text{N}$.

Explanation of Solution

Write the expression for the force of gravitation acting on the sail.

    $F_{\text{gravity}}=\frac{GMm}{r^2}$                                                    (I)

Here, $F_{\text{gravity}}$ is the force of gravitation, $G$ is the universal gravitational constant, $M$ is the mass of Earth, $m$ is the mass of sail and $r$ is the distance between Earth and sail.

Write the expression for the drag force on the sail.

    $F_{\text{drag}}=\frac{1}{2}C\rho v^{2}A$                                                 (II)

Here, $F_{\text{drag}}$ is the drag force, $C$ is the drag coefficient, $\rho$ is the density of Earth’s atmosphere, $v$ is the orbital velocity of sail and $A$ is the area of sail.

Write the expression for the orbital velocity of the sail.

    $v=\sqrt{\frac{G\cdot M}{R+r}}$                                                       (III)

Here, $G$ is the universal gravitational constant, $M$ is the mass of Earth, $R$ is the radius of Earth and $r$ is the altitude.

Write the value for distance between Earth and sail.

    $r=\left(6400\text{km}+\text{6}\text{.5}\times {10}^5\text{m}\right)$

Write the expression for the force exerted by the solar radiations on the sail.

    $F_{\text{radiation}}=\frac{2IA}{c}$                                                   (IV)

Here, $F_{\text{radiation}}$ is the force exerted by the radiations on the sail, $I$ is the intensity of the radiations, $A$ is the area of the sail and $c$ is the speed of light.

Conclusion:

Substitute $6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^{\text{2}}/{\text{kg}}^{\text{2}}$ for $G$, $6\times {10}^{24}\text{kg}$ for $M$, $4\text{kg}$ for $m$ and $\left(6400\text{km}+\text{6}\text{.5}\times {10}^5\text{m}\right)$ for $r$ in equation (I) to find $F_{\text{gravity}}$.

    $\begin{array}{c}{F}_{\text{gravity}}=\frac{\left(6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^{\text{2}}/{\text{kg}}^{\text{2}}\right)\left(6\times {10}^{24}\text{kg}\right)\left(4\text{kg}\right)}{\left(6400\text{km}\times \left(\frac{{10}^3\text{m}}{1\text{km}}\right)+\text{6}\text{.5}\times {10}^5\text{m}\right)}\\ =32.21\text{N}\\ \approx 30\text{N}\end{array}$

Substitute $6.67\times {10}^{-11}\text{N}\text{.}{\text{m}}^{\text{2}}/{\text{kg}}^{\text{2}}$ for $G$, $6\times {10}^{24}\text{kg}$ for $M$, $6400\text{km}$ for $R$ and $6.5\times {10}^5\text{m}$ for $r$. in equation (III) to find $v$.

    $\begin{array}{c}v=\sqrt{\frac{\left(6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^{\text{2}}/{\text{kg}}^{\text{2}}\right)\left(6\times {10}^{24}\text{kg}\right)}{\left(\left(6400\text{km}\right)\left(\frac{{10}^3\text{m}}{1\text{km}}\right)+6.5\times {10}^5\text{m}\right)}}\\ =7534.32\text{m}/\text{s}\end{array}$

Substitute $1$ for $C$, $5\times {10}^{-14}\text{kg}/{\text{m}}^{\text{3}}$ for $\rho$, $10{\text{m}}^{\text{2}}$ for $A$ and $7534.32\text{m}/\text{s}$ for $v$ in equation (II) to find $F_{\text{drag}}$.

    $\begin{array}{c}{F}_{\text{drag}}=\frac{1}{2}\times 1\times \left(5\times {10}^{-14}\text{kg}/{\text{m}}^{\text{3}}\right){\left(7534.32\text{m}/{\text{s}}^{\text{2}}\right)}^2\left(10{\text{m}}^{\text{2}}\right)\\ =1.42\times {10}^{-5}\text{N}\\ \approx 1.4\times {10}^{-5}\text{N}\end{array}$

Substitute $1.4\text{kW}/{\text{m}}^{\text{2}}$ for $I$, $10{\text{m}}^{\text{2}}$ for $A$ and $3\times {10}^8\text{m}/\text{s}$ for $c$ in equation (IV) to find $F_{\text{radiation}}$.

    $\begin{array}{c}{F}_{\text{radiation}}=\frac{\left(2\right)\left(1.4\text{kW}\times \frac{{10}^3\text{W}}{1\text{kW}}\right)\left(10{\text{m}}^{\text{2}}\right)}{3.0\times {10}^8\text{m}/\text{s}}\\ =9.33\times {10}^{-5}\text{N}\\ \approx 9\times {10}^{-5}\text{N}\end{array}$

Therefore, the gravitational force acting on the sail is $30\text{N}$. The drag force acting on the sail is $1.4\times {10}^{-5}\text{N}$. The force exerted by solar radiations on the sail is $9\times {10}^{-5}\text{N}$.