Question 347: Gravitation

Question

The average density of the earth is [g is acceleration due to gravity]

Accepted Answer

To determine the relationship between the average density of the Earth and the acceleration due to gravity (g), we start with the formula for gravitational force:
$$ F = G \frac{m_1 m_2}{r^2} $$
Here, $$G$$ is the gravitational constant, $$m_1$$ and $$m_2$$ are the masses of two objects, and $$r$$ is the distance between their centers of mass. For an object near the Earth's surface, this force is also equal to the weight of the object, which is given by:
$$ F = m g $$
Setting these two expressions for the force equal to each other, we get:
$$ m g = G \frac{m M_{	ext{earth}}}{r^2} $$
Here, $$M_{	ext{earth}}$$ is the mass of the Earth, and $$r$$ (or $$R$$) is the radius of the Earth. Simplifying, we obtain the equation:
$$ g = G \frac{M_{	ext{earth}}}{R^2} $$
Now, the mass of the Earth can be expressed in terms of its volume and average density ( $$ho_{	ext{earth}}$$ ) as follows:
$$ M_{	ext{earth}} = ho_{	ext{earth}} \cdot V_{	ext{earth}} $$
Since the volume of a sphere is given by
$$ V_{	ext{earth}} = \frac{4}{3} \pi R^3 $$
we can substitute this into our expression for $$M_{	ext{earth}}$$ to get:
$$ M_{	ext{earth}} = ho_{	ext{earth}} \cdot \frac{4}{3} \pi R^3 $$
Substituting this back into our equation for $$g$$, we have:
$$ g = G \frac{ho_{	ext{earth}} \cdot \frac{4}{3} \pi R^3}{R^2} $$
Simplifying the equation, we get:
$$ g = \frac{4 \pi G}{3} ho_{	ext{earth}} R $$
From this equation, we can see that the average density of the Earth, $$ho_{	ext{earth}}$$, is directly proportional to $$g$$, i.e.,
$$ ho_{	ext{earth}} \propto g $$
Hence, the correct answer is:
Option B
directly proportional to $$\mathrm{g}$$.

Answer

inversely proportional to g$$^2$$.

Answer

directly proportional to $$\mathrm{g}$$.

Answer

inversely proportional to $$\mathrm{g}$$.

Answer

directly proportional to $$\mathrm{g}^2$$.

Gravitation

The average density of the earth is [g is acceleration due to gravity]

Choose the correct answer: