"There is force of gravitation between the planets and the sun. Then why dont the planets get slammed on the surface of the sun due to the attraction generated by the force of gravitation?"

• Insights.

When a body revolves around another body in a circular path with constant radius and with constant speed, there is a force acting radially inward (i.e., towards the centre of the circular path as the Sun in the solar system) called the centripetal force. You might have learnt Newton's second law states that any kind of external force on a body brings about an acceleration in the body. But the speed of the moving body is constant (velocity changes due to acceleration), so what does the force do to it?

The thing here is that you need to know first the difference between velocity and speed. Speed is a scalar quantity i.e., just a magnitude. Velocity, on the other hand, has a direction associated with it too. The centripetal acceleration due to the centripetal force on the moving body changes the velocity for sure but just the direction not the magnitude.

Now, don't think that centripetal force is the force which keeps the body moving in the circular path. There needs to be a certain external force other than this to balance the two forces out. This is where the gravitational force comes into play.

Now the interesting question that should be coming in your mind now (even I thought of this) is that if centripetal force is towards the Sun and gravitational force is towards the Sun, why don't these two add up and not cancel each other?

This is where you need to understand that force and many other quantities are dependent on the observer. Suppose you are applying a force of $1N$ on your friend and your friend does nothing, then he would experience a force of $1N$ only. But if your friend exerts a force of $1N$ towards you he would feel as if no force is acting on him and if he pulls you back with a force of $1N$, he would experience double the amount of force that you've applied on him i.e., $2N$.

So while we tackle problems related to force from an inertial frame of reference (no external force on the observer body), all of Newton's laws are valid and forces, as seen by you, are the only forces acting. But in a planetary revolution, the gravitational force of the Sun is acting on the planets and the planets also are under the effect of the centripetal force. So the planet is not an inertial frame of reference and Newton's laws are invalid currently. What should we do to make them work?

The answer is simple. Pseudoforce. The pseudoforce is an additional external force that is not really acting on the body by the virtue of anything but it is taken into account to make Newton's laws valid in a non-inertial frame of reference too. The pseudoforce acting on the body is exactly same in magnitude but opposite to the direction of the actual internal force. Only centripetal force is accounted for pseudoforce in this case because gravitational forces are considered as external forces while centripetal force is due to the virtue of the circular motion done by the moving planet or body.

Hence, in the observation of the planet, the centripetal force acting radially inward (towards the centre of the Sun) has a pseudoforce called centrifugal force acting radially outward (away from the centre of the Sun). On the frame of the planet, if the pseudoforce and gravitational forces are such that they cancel each other then the planet revolves around the Sun in a constant circular path.

Now think about why under different circumstances, your friend felt the different amount of forces?

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• Are there no forces acting on the planet?

No. There are two forces acting on the planet. First is the mutual gravitational force between the Sun and the planet and the other is the centripetal force which comes due to the circular motion of the planets. But in the frame of planets where these forces are acting, they cancel out each other and the planet experiences no net force (net means resultant of all forces).

Note that the frame of reference is important is deciding what forces are acting on a body, but if generally asked like this you can assume that the person asking the question is asking from a frame in which no external forces are acting (in reality no frame is completely inertial but on a large scale we can assume Earth to be inertial when talking about small objects and the Sun to be inertial when talking about planets).

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• How does this phenomenon happen?

When heavenly bodies (like planets) in the universe get attracted towards a larger mass, they acquire some speed while approaching the body. Not usually is it the case they directly go towards the centre of the larger mass but sometimes if things are very right (right speed, right distance, right gravitational force), they may pass close to the body and get caught back by the gravitational force. When this happens the planet spirals towards the centre of the larger mass having their own velocities. In the most right circumstances, the planet finds a suitable distance and speed when no net force is acting on it. This is when the planet finds its orbit.

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• The reality!

Generally, the planetary model of a circular orbit is most widely accepted for elementary physics because it is easier to understand and gives a clear insight into the way planets move. In reality, most planets follow an elliptical path around their Suns and not a circular path. The reason for this is the changing velocities of the planets when this whole phenomenon of orbit formation takes place. Due to higher and lower velocities, the planets fit themselves in an elliptical orbit in such a manner that when their speed is maximum, they tend to be the nearest from the Sun and when they have the minimum speed, they tend to be farthest away from the sun.

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• The math behind it all.

The centripetal force acting on a body of mass $m$ moving in a circle of radius $r$ with velocity $v$ is given by

$F_C = \dfrac{mv^2}{r}$

And the mutual gravitational force of attraction between the body of mass $m$ and another body of mass $M$ at a separation $r$ from it is given by

$F_G = G \dfrac{mM}{r^2}$

where $G = 6.67 \times 10^{-11} \text{Nm}^2{ kg}^{-2}$ is the universal gravitation constant.

When these two forces balance each other, we have

$F_C = F_G \implies \dfrac{mv^2}{r} = G \dfrac{mM}{r^2} \implies v = \sqrt{\dfrac{GM}{r}}$

This is the orbital velocity of the planet, i.e., if the velocity is equal to this expression then the planet will not experience any net force on it and move in its orbit.

Note that for a planet-sun system $M$ is constant and $m$ is not a part of expression of $v$ and thus $v \propto \dfrac{1}{\sqrt{r}}$. Thus planets follow elliptical paths as mentioned before due to difference in velocities at different points of their orbital path.

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Feel free to ask your doubts in the reply section. Hope this helped.

What kind of force would it take to keep one body in uniform circular orbit around another stationary body (meaning constant orbital speed and constant orbital radius)? The answer is that a force in the inward radial direction is required. As it turns out, the gravitational attraction between the two bodies supplies exactly this kind of force. So uniform circular motion is perfectly consistent with mutual gravitation, as long as the parameters (mass, speed, etc.) are balanced properly.