Going round a circle was found to require an acceleration pointing at the centre. Where there is an acceleration there is a force. Writing the equation of motion toward the centre says the resultant toward the centre must be .
That resultant is called the centripetal force. It is a name. There is no new kind of force of size arriving from somewhere apart from gravity and tension.
This is where things get tangled. Having counted the forces on the body, one is tempted to draw in a further arrow labelled centripetal force. That is putting the resultant back among its own components, counting the same thing twice. The arrow in the figure is the tension in the string itself, not the tension with something added to it.
The left side of the equation is the resultant toward the centre, assembled from the forces that actually act; the right side is . Never write on the left.
So what is making the resultant point at the centre? It depends on the case.
Whirl the body on a string and it is the tension . Run it round the inside of a bowl or a cylinder and it is the normal force pressing inward. Orbit a heavenly body and it is gravitation . The figure exchanges these three in turn, and the way the body circles does not change. Each does the same job toward the centre; only the name and the source differ.
So the procedure is always the same. Count the forces on the body. Add up the components of those forces that point at the centre. Set the total equal to . That order is the same whether the partner is a string, a wall or gravity.
A single force is not always the whole of it. On a slope, or at the bottom of a dip on a coaster, the normal force and a part of gravity add together to make the resultant toward the centre. That is why the forces are counted first and the centripetal force settled afterward.
Cut the string of a body being whirled around. Which way does it fly?
It is not flung outward. It carries on straight along the tangent, with the velocity it had at the instant of the cut. This is the law of inertia exactly as stated: with no force acting, motion is uniform and in a straight line. What had been bending it was the tension, and with that gone it simply goes on without bending.
It looks as though it were flung outward because the body travelling straight and the circle curving away part company very quickly. The body was not pushed outward; the circle fell away inward.
The hammer throw and the drops flying off a spun umbrella are the same. Follow them with your eye after release and they do not scatter radially outward; each goes off along the tangent it happened to be on.
Speaking of outward forces, there is the centrifugal force. But it must never be drawn into a diagram taken from outside.
Every figure so far has been the view from outside. The only force acting is the tension in the string, and the body turns because the resultant points at the centre. What would happen if an outward force of the same size were added? The resultant would be and the body would travel straight. It plainly does not, so such a figure is wrong.
The figure in this scene alone is watched from a frame turning with the body. The body is at rest, and what appears to move is the marker outside the orbit, the surrounding scenery. Being at rest with only the tension acting makes no sense, so an outward force is added to balance it. That is the centrifugal force, of size , and the equation reads .
The centrifugal force has nothing on the other end of it. Search for the partner in an action and reaction pair and there is none; it appears only in the rotating view. A force of this kind is called an inertial force. Solve from outside or solve while turning; the answer is the same either way. The one thing not permitted is to mix the two views.