Centripetal acceleration v^2/r?

AI Thread Summary
Centripetal acceleration is defined as v^2/r, where v represents the velocity of an object moving in a circular path and r is the radius of that path. This relationship indicates that acceleration is proportional to the square of the velocity and inversely proportional to the radius of curvature. When an object changes direction, it experiences acceleration even if its speed remains constant, as acceleration is defined as the rate of change of velocity. The discussion highlights how inertia plays a role in this acceleration, with greater speeds and tighter turns resulting in more noticeable centrifugal forces. Understanding this relationship is essential for grasping the dynamics of circular motion.
hellbike
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can someone justify why centripetal acceleration = v^2/r?

And I'm not asking about algebraic proof.
 
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Would you like a calculus proof?

Let me look it up... nah...

Type "khanacademy calculus proof a=v^2/r" into Google. Then click the first video result.
 
hellbike said:
can someone justify why centripetal acceleration = v^2/r?

And I'm not asking about algebraic proof.

An object moving in space will move in a straight line at a constant velocity by Newton's first law of inertia. If the object is constrained in someway not to move in a straight line, then it must be experiencing an acceleration. By definition acceleration is a time rate change of velocity. If you don't want a mathematical proof of this, you must accept intuitively that this acceleration is not proportional to the velocity of the object, but is proportional to the velocity squared. Also it is inversely proportional to the radius of curvature of the object as it curves through space. Remember by definition acceleration is a time rate change of velocity. Velocity is a vector, so it has both magnitude and direction. If the object's direction is changing, then it must be accelerating also, even if the magnitude of the velocity is not. Imagine a car going around a circular race track with a radius of 100 feet at 65 miles per hour. It will experience an acceleration proportional to its speed squared and inversely proportional to its radius. The driver would experience an appreciable centrifugal force pulling him radially outward. Now imagine the same car going around a circular race track with the same speed, but the radius is now 100 miles. The acceleration will now be less. The driver would barely perceive the centrifugal force in his frame pulling him radially outward. It all boils down to the speed squared and how fast the car is turning in inertial space. Nature does not like change. For some unknown reason that is still not fully understood, inertia rises whenever there is a change in the velocity of an object. No one has yet come up with a fully accepted, bona fide explanation of the cause of inertial forces.
 
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