Question 321: Gravitation

Question

If two planets have their radii in the ratio $$x: y$$ and densities in the ratio $$m: n$$, then the acceleration due to gravity on them are in the ratio

Accepted Answer

Acceleration due to gravity of planet $$A$$ is given below.
$$g_A=\frac{G M_A}{R_A^2}$$ ..... (i)
$$\because$$ We know that, (Mass $$=$$ Density $$	imes$$ Volume) $$(M=D 	imes V)$$
So, from Eq. (i)
$$\begin{aligned}
g_A & =\frac{G D_A 	imes \frac{4}{3} \pi R_A^3}{R_A^2}=G D_A 	imes \frac{4}{3} \pi R_A \
g_A & =\frac{4}{3} \pi G R_A D_A \quad 	ext{.... (ii)}
\end{aligned}$$
Similarly, Acceleration due to gravity of planet $$B$$ is
$$g_B=\frac{G M_B}{R_B^2}$$ .... (iii)
$$	ext { and } \quad g_B=\frac{4}{3} \pi G D_B R_B$$ .... (iv)
We have to find the ratio of $$g_A: g_B$$
So, according to question, we have
$$\begin{aligned}
& \frac{R_A}{R_B}=\frac{x}{y} \quad 	ext{.... (v)}\
	ext{and} \quad & \frac{D_A}{D_B}=\frac{m}{n} \quad 	ext{.... (vi)}
\end{aligned}$$
Now, taking the ratio of Eqs. (ii) and (iv), we get
$$\frac{g_A}{g_B}=\frac{R_A D_A}{R_B D_B}$$
Put the value from Eqs. (v) and (vi) in this equation, we get
$$\frac{g_A}{g_B}=\frac{x \cdot m}{y \cdot n}$$
or $$\quad g_A / g_B=m x / n y$$

Answer

$$\frac{n y}{m x}$$

Answer

$$\frac{m y}{n x}$$

Answer

$$\frac{n x}{m y}$$

Answer

$$\frac{m x}{n y}$$

Gravitation

If two planets have their radii in the ratio $x: y and densities in the ratio m: n$, then the acceleration due to gravity on them are in the ratio

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