Acceleration due to gravity and its variation with height

Suppose a person reaches a height equal to half of radius of earth from its surface.

In that case, the person might feel that he doesn't have any weight because acceleration due to gravity becomes zero there.

Let's see the reason behind this.

For this, first, we shall find the acceleration due to gravity of earth at surface.

We will use Newton's second law of motion.

It says that the force acting on a body of mass $m$ moving with acceleration $a$ is given by,

In case of gravitational force, we will replace $a$ by acceleration due to gravity $g$.

Now, we will use Newton's law of gravitation in which the gravity force, $F$ is given by,

If we equate the two forces, since, both are same. We will get the value of $g$.

As we have calculated the value of $g$. Now we will proceed to find the variation of $g$ with height from the earth's surface.

So, let's try to find its variation with height.

As we know that the shape of earth is elliptical and not spherical. We will assume it to be spherical for calculation purpose.

When a body of mass $m$ is at the surface of earth, the acceleration due to gravity $g$ is given by,

Now, suppose that the body is at a height of $h$ from the earth's surface.

In this case, the distance between the center of earth and the position of the body is given by $R+h$.

Hence, the acceleration due to gravity acting on the body changes to $g^{\prime }$ which is given by,

Now, we will find the ratio of $g^{\prime }$ to $g$.

We need to simplify the obtained value of $g^{\prime }$.

Let's assume that $h$ is much smaller than $R$ and so, we will neglect the higher order of $\frac{h}{R}$ in binomial expansion since, they tend to become zero.

So, applying the Bernoulli's expansion will give the expression of $g^{\prime }$.

So, we can see that $g$ acting on a body decreases when taken to a certain height $h$ from surface of earth.

Somehow, if we are able to reach a height of $\frac{R}{2}$, then, the acceleration due to gravity reduces to zero there.

Revision

The acceleration due to gravity $g$ varies with height $h$ as shown,

From the expression, we can analyze that the acceleration due to gravity decreases with increase in height from the surface of earth.

The End