Centripetal Acceleration
A body moving in a circle (or curve path in general) has a linear velocity that is changing with time because its direction is changing all the time. THerfore, from the definition of acceleration as rate of change of velocity, that means, it has non-zero acceleration.
For a body moving in a circle, this acceleration is points towards the center of the circle. Therefoer, it is called centrpetal acceleration (frmo Latin: center= "enter", and petere = "to seek").
Centripetal acceleration must not be confused with angular acceleration, which refers to rate of change of angular velocity.
Centripetal acceleration is always written in terms of the radius of the circular path, \(r\), and either the tangential velocity, \(v\), or angular velocity, \(\omega\):
\[a_c=\frac{v^2}{r}=\omega^2 r\]
Centripetal Force
A body in circular motion experience centripetal acceleration pointing towards the center of the circular trajectory. Therefore, by Newton:s second law, there must be a resultant force on it that is also directed towards the center. This is known as centripetal force.
Centripetal force is the force that is needed to keep an body in circular motion.
\[F=\frac{mv^2}{r}=mr\omega^2\]
In vector form, the first formula can be written as follows:
\[F=-\frac{mv^2}{r^2}\vec r\]
The minus sign arises because the force points towards the center (opposite to the \(\vec r\) which is pointing outwards.
Useful References
The minus sign arises because the force points towards the center (opposite to the \(\vec r\) which is pointing outwards.
Useful References
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