Question

Choose the correct alternative:
A. Acceleration due to gravity increases/decreases with increasing height.
B. Acceleration due to gravity increases/decreases with increasing depth. (Assume earth to be a sphere of uniform density)
C. Acceleration due to gravity is independent of mass of the earth/mass of the object.
D. The formula $-\text{GMm}\left( \text{1/}{{\text{r}}_{\text{2}}}-\text{1/}{{\text{r}}_{\text{1}}} \right)$ is more/less accurate than the formula $\text{mg(}{{\text{r}}_{\text{2}}}-{{\text{r}}_{\text{1}}}\text{)}$for the potential energy between two points ${{\text{r}}_{1}}\text{ and }{{\text{r}}_{2}}$ distance away from the earth.

Answer 1

Hint: The acceleration due to gravity varies as we go up or down the surface of earth. Consider what happens when we go very high from earth and reach a point in outer space. What happens to due acceleration due to gravity at that time. Also consider the equation of acceleration to gravity and look at how g is related to each term.

Complete step by step answer:
The acceleration due to gravity is the acceleration attained by a body in vacuum while it is free falling or falling under gravity alone. This acceleration is caused by the gravitational attraction of the planet and the body. The acceleration due to gravity depends on the mass of the planet where the object is dropped and the distance between the plane and the object. It is denoted by ‘g’. The acceleration due to gravity for a planet of mass M and radius R is given by,
$\text{g}=\dfrac{\text{GM}}{{{\text{R}}^{\text{2}}}}$
Where, G is the gravitational constant.
From the equation it is clear that the acceleration due to gravity is dependent on the mass and radius of the planet.
The above equation gives the acceleration due to gravity value at the surface of the earth. So what happens if we take the object up or down the surface of the earth?
The change in acceleration due to gravity at a height h above the surface of the earth is given by,
${{g}_{h}}=\dfrac{g}{{{\left( 1+\dfrac{h}{R} \right)}^{2}}}$
Where,
${{\text{g}}_{\text{h}}}$ is the acceleration due to gravity at a height ‘h’ from earth.
g is the acceleration due to gravity at the surface of the earth.
From the equation it is clear that as h increases the value of g decreases.

The value of g at a depth ‘d’ (assuming that the earth has uniform density) from the surface of the earth is given by the formula
${{g}_{d}}=g\left[ 1-\dfrac{d}{R} \right]$
Where,
${{\text{g}}_{\text{d}}}$ is the acceleration due to gravity at a depth ‘d’ from earth.
g is the acceleration due to gravity at the surface of the earth.
So the value of g decreases as we go to a depth from the surface of the earth. So the maximum value of g is at the surface of the earth.
The gravitational potential energy at a distance r from earth is given by the formula,
$\text{U}=-\dfrac{\text{GMm}}{\text{r}}$
So the difference in potential energy between two points which is at a distance ${{\text{r}}_{1}}\text{ and }{{\text{r}}_{2}}$ away from the earth is given by,
$-\text{GMm}\left( \text{1/}{{\text{r}}_{\text{2}}}-\text{1/}{{\text{r}}_{\text{1}}} \right)$
This equation is better than the potential energy equation $\text{mg(}{{\text{r}}_{\text{2}}}-{{\text{r}}_{\text{1}}}\text{)}$for the potential energy between two points ${{\text{r}}_{1}}\text{ and }{{\text{r}}_{2}}$ which is applicable at small heights near the surface of the earth.

So the answers to the questions are:
Acceleration due to gravity decreases with increasing height.
Acceleration due to gravity decreases with increasing depth. (Assume earth to be a sphere of uniform density)
Acceleration due to gravity is independent of mass of the object.
The formula $-\text{GMm}\left( \text{1/}{{\text{r}}_{\text{2}}}-\text{1/}{{\text{r}}_{\text{1}}} \right)$ is more accurate than the formula $\text{mg(}{{\text{r}}_{\text{2}}}-{{\text{r}}_{\text{1}}}\text{)}$for the potential energy between two points ${{\text{r}}_{1}}\text{ and }{{\text{r}}_{2}}$ distance away from the earth.

Note: The value of acceleration due to gravity at the center of the earth is zero.
The object is said to be at infinity if acceleration due to gravity on the object above the surface of the earth is zero.