Question

Figure 13-27 gives the gravitational acceleration ${\mathit{a}}_{\mathbf{g}}$for four planets as a function of the radial distance rfrom the center of the planet, starting at the surface of the planet (at radius${\mathit{R}}_{\mathbf{1}}\mathbf{,}{\mathit{R}}_{\mathbf{2}}\mathbf{,}{\mathit{R}}_{\mathbf{3}}\mathbf{}\mathit{o}\mathit{r}\mathit{}{\mathit{R}}_{\mathbf{4}}$ ). Plots 1 and 2 coincide for $\mathit{r}\mathbf{\ge }{\mathit{R}}_{\mathbf{2}}$plots 3 and 4 coincide for$\mathit{r}\mathbf{\ge }{\mathit{R}}_{\mathbf{4}}$.Rank the four planets according to (a) mass and (b) mass per unit volume, greatest first.

Short answer

1. The order of the planets according to mass is 1 = 2 > 3 = 4.
2. The order of the planets according to mass per unit volume is 1 > 2 > 3 > 4.

Step-by-step solution

The given data

1. The graph of gravitational acceleration $a_g$versus radius of four planets is given.
2. Plots 1 and 2 coincide for $r\ge R_2$.
3. Plots 3 and 4 coincide for $r\ge R_4$.

Understanding the concept of the force

The force of attraction between any two bodies is directly proportional to the product of their masses and is inversely proportional to the square of the distance between them, according to Newton's universal law of gravitation.We can find the relation between mass and gravitational acceleration by equating the gravitational force of attraction and the gravitational force by Newton’s second law. Hence, we can rank the planets according to their mass. Then, using the relation between mass and density (mass per unit volume) we can rank them according to their density.

Formulae:

The Gravitational force of attraction between two bodies of masses M and m separated by distance R is,$F=\frac{GMm}{R^2}$ …(i)

The force on a body due to Newton’s second law,$F=ma$ …(ii)

The density of a body, $p=\frac{M}{V}$ …(iii)

(a) Calculation of the order of the planets according to the mass

Using the equation (ii), we can get the value of the force due to the gravitational acceleration as follows:

$F_g=m{a}_g$

Now, comparing the force value with the gravitational force of equation (i) for distance r we get the equation as follows:

$$\begin{gathered} m{a}_g=\frac{GMm}{r^2} \\ {a}_g=\frac{GM}{r^2} \\ M=\frac{a_g{r}^2}{G} \end{gathered}$$

It interprets that the mass of the planet is directly proportional to gravitational acceleration and the square of the distance. So, applying this to the given graph we can conclude that planet 1 and 2 have greater mass than planet 3 and 4.

Since graph for 1 coincides with that for 2, and graph for 3 coincides with that for 4, the masses of 1 and 2 are the same. Also, the masses of 3 and 4 are the same.

Therefore, the order of the planets according to mass is 1 = 2 > 3 = 4.

(b) Calculation of the order of the planets according to the mass per volume

If we consider planet is a sphere, the n mass per unit volume that is density can be given using equation (iii) as:

$p=\frac{M}{{\displaystyle \frac{4}{3}}{\mathrm{\pi r}}^3}(\mathrm{Volume}\mathrm{of}\mathrm{the}\mathrm{sphere}=\frac{4}{3}{\mathrm{\pi r}}^3)$

This shows that density is directly proportional to mass. So, planets 1 and 2 have greater density than planets 3 and 4.

It is also inversely proportional to the radius of the planet. So, the planets having greater radii will have lesser density.

Since the radius of planet 2 is greater than the radius of planet 1, planet 1 will have greater density than that of planet 2. Similarly, planet 3 will have greater density than that of planet 4.

Therefore, the order of the planets according to mass per unit volume is 1 > 2 > 3 > 4.