# Linear Velocity and Acceleration

## Main Question or Discussion Point

Hey guys, we have just finished Chapter 10 (Rotation) and I have some questions regarding some of the concepts. For an object to rotate about some axis, any given particle at point P at some radius "r" has a linear velocity (tangential velocity) and linear acceleration (tangential acceleration). Now, I know that for a particle/object undergoing uniform circular motion, it has a velocity tangential to is radial acceleration. However, how come when you have it rotating about a fixed axis, it has both a tangential acceleration and radial acceleration? Furthermore, they say that the acceleration of the object is the magnitude of both tangential and radial acceleration? Is this magnitude "a" supposed to be the actual acceleration of the particle/object if it was traveling in a linear motion? Or is it because of the fact that a = delta v / delta t?

Thanks.

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There is only tangential acceleration if the rotation is speeding up (if there is radial acceleration). However there is always centripetal acceleration if an object is rotating. You can imagine that as a string holding a ball being swung around. The centripetal acceleration is keeping the ball going in a circle. If the string broke suddenly, it would only have a velocity tangential to where it broke.

Now if there is tangential and centripetal acceleration, the actual acceleration of the object is a=$$\large\sqrt{a^{2}_{c}+a^{2}_{t}}$$. This is because the tangential and centripetal accelerations form a 90 degree angle. This is the linear acceleration (dv/dt), and the actual acceleration. These things are all different ways of stating the same thing. It is the acceleration of the object.

This wikimedia photo might help conceptually This is if there is no angular acceleration. That is, tangential velocity is constant and nonzero.