Question

If a graph is plotted between $t^2$ and $r^3$ for a planet then its slope will be ____________

Answer 1

Step 1: Given Data

Orbital radius $=r$

Time period $=t$

Let $v_0$ be the orbital velocity.

Let $G$ be the universal gravitational constant.

Let $M$ be the mass of the planet.

Let $m$ be the slope of the graph.

Let $c$ be the intercept of the graph.

Step 2: Formula used

We know that orbital velocity is given as,

$v_0=\sqrt{\frac{GM}{r}}$

We know that the time period is given as,

$t=\frac{2\mathrm{\pi r}}{v_0}$

$\Rightarrow t=\frac{2{\mathrm{\pi r}}^{{\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}{\sqrt{GM}}$

$\Rightarrow t^2=\frac{4{\mathrm{\pi }}^2}{GM}{r}^3$ $\left(1\right)$

Step 3: Calculate the Slope

The equation of a straight line is given as,

$y=mx+c$ $\left(2\right)$

Upon comparing equation $\left(1\right)$ and equation $\left(2\right)$ we get,

$m=\frac{4{\mathrm{\pi }}^2}{GM}$

Step 4: Graph

Hence, if a graph is plotted between $t^2$ and $r^3$ for a planet then its slope will be $\frac{4{\mathrm{\pi }}^2}{GM}$.