What is the difference between centripetal acceleration and acceleration due to gravity?

The acceleration due to gravity really is just what it says, namely the acceleration that a mass will experience simply by being within the gravitational field of the Earth. The force applied by gravity however is not quite the same; it is equal to the mass of the object multiplied by its acceleration. So that's the key difference; the force multiplies the acceleration by the mass of the object.

This property is laid out by Newton's Second Law, given by the formula F=ma where F is the force, m is the mass and a is the acceleration. Force is, if you like, a description of how quickly the momentum of an object is changing and so it kind of makes sense that its all about pushes and pulls. For example when a tennis player hits the ball on serve you'lllll agree intuitively that the momentum of the ball is increased; the ball is 'pushed' forward.

You are completely correct in saying weight is equal to the force due to gravity; they are just two interchangeable phrases, both are equally ... more.

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