Proof of the equation of centripretal force

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The discussion focuses on the complexities of centripetal force in circular motion compared to projectile motion, highlighting the challenges of applying the parallelogram law when dealing with uniform velocity and centripetal acceleration. It notes that while projectile motion simplifies the analysis by assuming constant downward acceleration, circular motion introduces confusion due to simultaneous acceleration toward the center. A geometric proof of centripetal force is mentioned as being more intuitive than a vectorial approach. Participants debate the applicability of the parallelogram law in circular motion, suggesting that it can still be utilized by considering small angles. The conversation emphasizes the need for clarity in understanding these concepts in physics.
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When we apply the parallelogram on two velocities acting on a single body we see that the velocity in anyone of the two directions remains the same. ( the velocities act on different directions)

When one is velocity and the other acceleration or in other words accelerating a body in some direction other than the body's uniform velocity the situation gets complex. In projectile motion we assume the acceleration to act downwards (rather than the centre) in all the points of its trajectory so that the downward forces are parallel in all the points of its trajectory. This makes the problem easy. And it is legitimate considering the size of earth.

But when we consider circular motion, that's when i get confused. Because here we are dealing with bodies moving in a uniform velocity and at the same time accelerating toward the centre. SO i don't know how to manipulate such conditions. Here the parallelogram law is not applied. I have seen a proof of the equation of centripretal force using pure geometry. And it is intuitive than the vectorial proof. Can it be done for motion in a vertical circle? or is calculus necessary?
 
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hi batballbat! :smile:
batballbat said:
But when we consider circular motion, … Here the parallelogram law is not applied.

yes it is!

consider the velocities at small angles ±θ from some direction …

they're equal in magnitude, so draw two lines of equal length from the same point, at angles ±θ …

the line joining them, to make the diagonal of the parallelogram, is at 90° - θ to both velocities :wink:
 

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