91Ó°ÊÓ

The given data can be listed below as,

• The radius of planet Uranus is,$R_u=25360\text{ k³¾}$.
• The surface acceleration due to gravity is, $g_u=9.0\text{″¾}/{\text{s}}^{\text{2}}$.
• The altitude of the circular orbit of moon Mirandais, $h=104000\text{ k³¾}$.
• The mass of Miranda is, $m_m=6.6×{10}^{19}\text{ k²µ}$.
• The radius of Miranda is,$r_m=236\text{ k³¾}$.

The attractive force between two objects is inversely proportional to the square of the distance between them and directly proportional to the product of their masses.

a)

The relation between gravitational force, the mass of Uranus, the mass of Miranda, and the radius of Uranus is expressed as,

$w=F=\frac{G{M}_u{m}_m}{R_u^2}$ ...(i)

Here$F$ is the gravitational force, $w$is the weight of a body at the planet surface,$G$is the gravitational constant, $M_u$is mass of Uranus,$m_m$is mass of Miranda, and$R_U$is the radius of Uranus.

The relation between the weight of the body, mass of body, and acceleration of planet is expressed as,

${\text{w=m}}_m{g}_u$

Here$g_u$is the acceleration due to gravity at its poles.

Substitute the value of $w$in equation (i)

$\begin{array}{c}{m}_m{g}_u=\frac{G{M}_u{m}_m}{R_u^2}\\ g_u=\frac{G{M}_u}{R_u^2}\\ M_u=\frac{g_u{R}_u^2}{G}\end{array}$

Substitute$9.0\text{″¾}/{\text{s}}^{\text{2}}$ for$g_u$ ,$2.5360×{10}^7\text{″¾}$ for $R_u$, and$6.673×{10}^{−11}{\text{ N³¾}}^{\text{2}}/{\text{kg}}^{\text{2}}$ for$G$ in the above equation.

$\begin{array}{c}{M}_u=\frac{9.0\text{″¾}/{\text{s}}^{\text{2}}×{(2.5360×{10}^7)}^2\text{″¾}}{6.673×{10}^{−11}{\text{ N³¾}}^{\text{2}}/{\text{kg}}^{\text{2}}}\\ =8.674×{10}^{25}\text{ k²µ}\end{array}$

Hence the mass of Uranus is,$8.674×{10}^{25}\text{ k²µ}$ .

b)

The relation between gravitational force, mass of Uranus, mass of Miranda and radius of Uranus is expressed as,

$w=F=\frac{G{M}_u{m}_m}{r^2}$ ...(ii)

Here$r$is the orbital radius of Miranda. It is expressed as,

$r=h+R_u$

The relation between weight of body, mass of body and acceleration of Miranda is expressed as,

${\text{w=m}}_m{g}_m$

Here$g_m$is the acceleration of Miranda.

Substitute the value of $w$and$r$in the equation (ii).

$\begin{array}{c}{m}_m{g}_m=\frac{G{M}_u{m}_m}{{(h+R_u)}^2}\\ g_m=\frac{G{M}_u}{{(h+R_u)}^2}\end{array}$

Substitute the value of$M_u$in the above equation.

$\begin{array}{c}{g}_m=\frac{G\left(\frac{g_u{R}_u^2}{G}\right)}{{(h+R_u)}^2}\\ g_m=\frac{g_u{R}_u^2}{{(h+R_u)}^2}\end{array}$

Substitute$9.0\text{″¾}/{\text{s}}^{\text{2}}$ for $g_u$, $2.5360×{10}^7\text{″¾}$for $R_u$, and $1.04×{10}^8\text{″¾}$ for$h$ in the above equation.

$\begin{array}{c}{g}_m=\frac{9.0\text{″¾}/{\text{s}}^{\text{2}}×{(2.5360×{10}^7\text{″¾})}^2}{{(1.04×{10}^8\text{″¾}+2.5360×{10}^7\text{″¾})}^2}\\ =0.346\text{″¾}/{\text{s}}^{\text{2}}\end{array}$

Hence the acceleration of Miranda due to its orbital motion around Uranus is,$0.346\text{″¾}/{\text{s}}^{\text{2}}$ .

c)

The relation between acceleration mass of Miranda, gravitational constant and radius of Miranda is expressed as,

$g=\frac{G{m}_m}{r_m^2}$

Here g is the acceleration.

Substitute$6.673×{10}^{−11}{\text{ N³¾}}^{\text{2}}/{\text{kg}}^{\text{2}}$ for$G$ ,$6.6×{10}^{19}\text{ k²µ}$ for$m_m$ and$2.36×{10}^5\text{″¾}$ for$r_m$ in the above equation.

$\begin{array}{c}g=\frac{6.673×{10}^{−11}{\text{ N³¾}}^{\text{2}}/{\text{kg}}^{\text{2}}×6.6×{10}^{19}\text{ k²µ}}{{(2.36×{10}^5\text{″¾})}^2}\\ =0.079\text{″¾}/{\text{s}}^{\text{2}}\end{array}$

Hence the acceleration due to Miranda’s gravity at the surface of Miranda is,$0.079\text{″¾}/{\text{s}}^{\text{2}}$ .

d)

No. The object and Miranda orbit Uranus together due to the gravitational pull of Uranus. The entity has additional abilities with respect to Miranda.