A palindrome is a number or a text phrase that reads the same backwards as forwards. For example, each of the following five-digit integers is a palindrome: 12321, 55555, 45554 and 11611. Write a C++ program that reads in a five-digit integer and determines whether it is a palindrome .

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Textbook Solution:

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Answer:

To determine

Rank the energy quantities from largest to smallest.

Answer

The ranking is a>b=c−−−−−−−−$\underset{\_}{a>b=c}$ the energy quantities from largest to smallest.

Explanation

a$a$ The absolute value of the average potential energy of the sun-earth system.

b$b$ The average kinetic energy of the earth in its orbital motion relative to the sun.

c$c$ The absolute value of the total energy of the sun-earth system.

Write the expression for the potential energy between the sun-earth system.

    U=−GMSMER$U=-\frac{G{M}_S{M}_E}{R}$        I$I$

Here, U$U$ is the average potential energy of the sun-earth system, G$G$ is the gravitational constant, MS$M_S$ is the mass of sun, ME$M_E$ is the mass of earth, R$R$ is the separation between the sun and earth.

Use equation I$I$ to write the absolute value of U$U$.

    |U|=GMSMER$\left|U\right|=\frac{G{M}_S{M}_E}{R}$        II$II$

Write the expression for the kinetic energy.

    KE=12MEv2E$K_E=\frac{1}{2}{M}_E{v}_E^2$        III$III$

Here, KE$K_E$ is the kinetic energy of the earth, vE$v_E$ is the velocity of the earth.

Write the expression for the gravitational force.

    FG=GMEMSR2$F_G=\frac{G{M}_E{M}_S}{R^2}$        IV$IV$

Here, FG$F_G$ is the gravitational force.

Write the expression for the centripetal force of the earth.

    FC=MEv2ER$F_C=\frac{M_E{v}_E^2}{R}$        V$V$

Here, FC$F_C$ is the centripetal force.

Equate equation IV$IV$ and V$V$ to solve for v2E$v_E^2$.

    v2E=GMSR$v_E^2=\frac{G{M}_S}{R}$        VI$VI$

Use equation VI$VI$ in III$III$ to solve for KE$K_E$.

    KE=GMSME2R$K_E=\frac{G{M}_S{M}_E}{2R}$        VII$VII$

Write the expression for the total energy of the sun-earth system.

    E=KE+U$E=K_E+U$        VIII$VIII$

Here, E$E$ is the total energy of the earth-sun system.

Use equation I$I$ and VII$VII$ in VIII$VIII$ to solve for E$E$.

    E=GMSME2R−GMSMER=−GMSME2R$\begin{array}{c}E=\frac{G{M}_S{M}_E}{2R}-\frac{G{M}_S{M}_E}{R}\\ =-\frac{G{M}_S{M}_E}{2R}\end{array}$        IX$IX$

Use equation IX$IX$ and write the absolute value of E$E$.

    |E|=GMSME2R$\left|E\right|=\frac{G{M}_S{M}_E}{2R}$        X$X$

Conclusion:

The ranking of the energy quantities from largest to smallest is |U|>KE=|E|$\left|U\right|>K_E=\left|E\right|$.

Therefore, the ranking is a>b=c−−−−−−−−$\underset{\_}{a>b=c}$ the energy quantities from largest to smallest.