**Write an expression from the given data to calculate the mass of the planet concerning**G**and the variables given in the statement.****Also calculate the mass of the planet in**Kg**if**T=26**hours and**R=2.1X10^8m.

This problem aims to familiarize us with the **objects revolving** around a specific **pivot point.** The concepts required to solve this problem are mostly related to **centripetal force**, **centripetal acceleration** and **orbital velocity.**

According to the **definition, centripetal** **force** is the **force** acting on an object rotating in a **circular** orientation, and the object is **pulled** towards the axis of **rotation** also known as the center of **curvature.**

The formula for **Centripetal Force** is shown below:

\[ F = \dfrac{mv^2}{r}\]

Where $m$ is the **mass** of the object given in $Kg$, $v$ is the **tangential velocity** in $m/s^2$ and $r$ is the **distance** of the object from the **pivot** point such that if the **tangential velocity** doubles, the **centripetal force** will be increased four times.

Another term to be **aware** of is **orbital velocity,** which is the **velocity** fine enough to induce a **natural** or **unnatural** satellite to stay in **orbit.** Its formula is:

\[ V_{orbit} = \sqrt{\dfrac{GM}{R}}\]

Where $G$ is the **gravitational constant,**

$M$ is the **mass** of the body,

$R$ is the **radius.**

## Expert Answer

The information given in the problem statement is:

The **time period** of spaceship $T = 26\space hours$,

The **distance** of the spaceship from the axis $R = 2.1\times 10^8\space m$.

For finding the **general expression** for the mass of the planet, we will be using the formula of **centripetal gravitational force** because it provides the necessary **centripetal acceleration** as:

\[F_c=\dfrac{GMm}{R^2}………………..(1)\]

**Centripetal acceleration** is given as :

\[a_c = \dfrac{v^2}{R}\]

Also from **newtons second equation** of motion:

\[F_c = ma_c\]

\[F_c = m(\dfrac{v^2}{R})\]

**Substituting** the value of $F_c$ in equation $(1)$:

\[\dfrac{GMm}{R^2} = m (\dfrac{v^2}{R})\]

**Simplifying** the equation gives us:

\[v = \sqrt{\dfrac{GM}{R}}\]

Where $v$ is **orbital velocity,** also:

\[v = \dfrac{total\space distance}{time\space taken}\]

Since the total **distance** covered by the spaceship is **circular,** it will be $2\pi R$. This gives us:

\[v = \dfrac{2\pi R}{T}\]

\[\dfrac{2\pi R}{T} = \sqrt{\dfrac{GM}{R}}\]

**Squaring** on both sides:

\[(\dfrac{2\pi R}{T})^2 = (\sqrt{\dfrac{GM}{R}})^2\]

\[\dfrac{4\pi^2 R^2}{T^2} = \dfrac{GM}{R}\]

**Rearranging** it for $M$:

\[M = (\dfrac{4\pi^2}{G}) \dfrac{R^3}{T^2}\]

This is the **general expression** to find the **mass** of the planet.

Substituting the values in the above **equation** to find the **mass:**

\[M = (\dfrac{4\pi^2}{6.67\times 10^{-11}}) \dfrac{(2.1\times 10^8)^3}{(26\times 60\times 60)^2}\]

\[M = (\dfrac{365.2390\times 10^{24+11-4}}{6.67\times 876096})\]

\[M = 6.25\times 10^{26}\space kg\]

## Numerical Result

The **expression** is $M=(\dfrac{4\pi^2}{G}) \dfrac{R^3}{T^2}$ and the **mass** of the **planet** is $M=6.25\times 10^{26}\space kg$.

## Example

A $200 g$ **ball** is revolved in a **circle** with an **angular speed** of $5 rad/s$. If cord is $60 cm$ **long,** find $F_c$.

The equation for **centripetal force** is:

\[ F_c = ma_s \]

\[ F_c = m \dfrac{v^2}{r} = m \omega^2 r\]

Where $\omega$ is the **angular velocity,** substituting the values:

\[ F_c = 0.2\times 5^2\times 0.6 \]

\[ F_c = 3\space N \]