 # Gravity IIT JEE Study Material

The natural phenomenon by which all the objects with mass are brought towards one another, including objects like atoms, photons, stars, planets, etc is called gravity.It’s S.Iunit is m s -2.

## Gravitational Potential Energy of a System:

Change in gravitational potential energy U (r) of two point masses m1and m2as their separation is changed from r1to r2is given by,

$\mathbf{U\left ( r_{2} \right )\;-\;U\left ( r_{1} \right )\;=\;\frac{G\;m_{1}\;m_{2}}{r_{1}\;-\;r_{2}}}$

At infinite separation the gravitational potential energy (Ur) is assumed to be 0,

i.e. $\mathbf{U\left ( r \right )\;=\;-\;\frac{G\;m_{1}\;m_{2}}{r}}$

## Gravitational Potential (V)

The expression for the potential is given by

$\mathbf{V\;=\;\int_{\infty }^{P}\;\;\overrightarrow{E}\;.\;\overrightarrow{dr}}$

The gravitational potential due to a point mass M at a distance r from it is $\mathbf{V\;=\;\frac{GM}{r}}$

Potential is a scalar quantity. If masses m1, m2, m3, . . . . . . . . mnare at distances r1, r2, . . . . . . . , rnthen the potential at any given point is

$\mathbf{V\;=\;-\;G\;\left [ \frac{m_{1}}{r_{1}}\;+\;\frac{m_{2}}{r_{2}} \;+\;.\;.\;.\;.\;.\;\right ]}$

The potential and the field are related as $\mathbf{E\;=\;-\;\frac{dV}{dr}}$

## Gravitational Potential (V) due to Uniform Solid Sphere:

A. At distance r from the center (Outside the sphere)

$\mathbf{V\;=\;-\;\frac{GM}{r}}$

B.At distance r from the center(Inside the sphere)

$\mathbf{V\;=\;-\;\frac{GM}{R^{3}}\;\left [ \frac{R^{2}}{2} \;-\;\frac{r^{2}}{6} \right ]}$

## Escape Velocity:

The escape velocity is the minimum velocity given to an object to make it free from the gravitational field. Let vebe the escape velocity of the body and m be the mass of the object to be projected then,

$\mathbf{v_{e}\;=\;\sqrt{\frac{2\;G\;M}{R}}}$

Don’t you hate falling objects, especially when they land on you?God forbid a crow flies over your head and suddenly decides to take a dump. Or worse yet, a much heavier object falls over you, well then you’ll meet your maker soon enough. But fear not, if nothing else I shall tell you where to point your finger or whom or rather what to blame, gravity. We, as a race have come far, from rubbing two stones together, watching the sparks and the resultant fire with awe and curiosity to building robots. And somewhere in between, a great man called Newton watched an apple fall from a tree and rather than picking it up and eating it as most of us would have, he wondered what invisible forces drive the apple to fall down, rather than up or towards the \left or so on. This is the same kind of curiosity that the human spirit boasts about\; this is the same curiosity that led to the discovery of penicillin and the other great inventions in human history. You see, the force of gravity is merciless. We’ve all felt it, it begins with us losing balance, and before you know it, you’ve toppled over and lying down bruised. That right there is thanks to the gravitational force. Science has described it as a certain attractive force exerted by all the objects in the universe. They say that every single object in the universe attracts every other object with a force directed along the line of centers for the two objects that is proportional to the product of their masses and inversely proportional to the square of the separation between the two objects. Scientists are known to be obstinate creatures. These very scientists set out on a journey a long time back, to comprehend exactly what gravity entails. That journey has still not reached its destination. There’s still only so much we know about gravity. Let’s go through what we do know about it.

Gravity does not just exist on earth. Most of the movies we’ve seen misdirect us, in the sense, we are made to feel that gravity does not exist on the moon. This is false. Gravity exists everywhere. But the moon just does not has as strong gravity as the earth. One might wonder if this force of gravity is all prevalent, then why don’t the chairs and tables jump at each other? This is because the force of gravity of earth is much stronger than the chair and table’s force of gravity. It also depends on how far away you are from this force of gravity. The farther away you go from the earth, the less the force of gravity affects you.

So you see, humans have accomplished something extraordinary. We were never meant to fly. But we overcame gravity, built flights and aircraft to soar the skies. Now every time you catch an apple that’s about to fall, you know that you’ve defied gravity.