What is the difference between angular acceleration and centripetal acceleration?

Question

One is defined as

α = Δω / Δt

The other as:

a = v^2/r

Whats the difference in a more conceptual theoretical form?

Answer 1

The 1st expression refers to the angular acceleration (degrees/second/second), independent of the radius.

The 2nd expression refers to the acceleration at a given radius.

The same angular acceleration will vary at different radii.

Answer 2

Imagine you have a ball attached to the end of a string. And you start spinning that ball above your head in fast circles.

If you keep the speed of the ball constant your angular acceleration is zero, but you have a centripetal acceleration that's perpendicular to the motion of the ball.

However, if you are continually spinning the ball faster and faster, then you still have an centripetal acceleration, but you also have a positive angular acceleration (which tells you how fast you are speeding up the rotation).

And like wise if you start slowing down the ball you will have a negative angular acceleration.

Answer 3

Angular Acceleration:

If you have the turntable going at the 33 1/3 RPM speed and switch it to 45 RPM, the increase in rotation speed is due to Angular Acceleration. It acts on all atoms making up the record and the direction of the vector is along the axis of rotation. The polarity is according to the "Right Hand Rule" - holding your right hand over the turntable with your fingers following the increasing angular velocity, your thumb points in the direction of the vector. I don't remember if turntables rotate Clockwise (CW) or Counter-Clockwise (CCW). Let's say it's CW, then the acceleration vector points down into the plane of the record. If you change the speed from 45 to 33 1/3, the acceleration vector would point up because it's a deceleration.

  Centripetal Acceleration:

If you put a coin at the outer edge of the record, and if it doesn't fly off, it's because there is enough friction to hold it. Assume the turntable is at a constant speed setting. The linear velocity vector of the coin has a constant magnitude, but its direction is continually changing. At all points, the velocity vector is tangential to the circular path it follows.

In order to change a velocity vector, either in magnitude or direction, acceleration must be applied. In the case of our coin, centripetal acceleration is what it is called - it means seeking a center. The value of this acceleration is given by (v^2)/r where v is the linear velocity of the coin and r is its distance from the center. Friction is providing the force for the centripetal acceleration in the case of our coin. Satellites orbiting the Earth also experience centripetal acceleration, provided by gravity.

The term Centripetal Force is also used in these discussions. It's whatever force is providing the centripetal acceleration. In the case of our coin, the force of friction is the Centripetal Force.

Circular Acceleration: I wasn't familiar with this term. Go to the site http://imagine.gsfc.nasa.gov/YBA/cyg-X1-... and note the comments to the right of the animation. It seems that this term is synonymous to centripetal acceleration.

Centripetal Force, OK what's Centrifugal Force?:

Newton's 3rd law says that a reaction force must counter the force for the centripetal acceleration. The centrifugal force is often mistakenly thought to cause a body to fly out of its circular path when it is released; but, it is the removal of the centripetal force that allows the body to travel in a straight line as required by Newton's first law. Centrifugal force is a virtual, or phantom, force.  It is not really a force. Think of a bag of groceries sitting on the seat of a car when the driver slams on the brakes. Does some force make the groceries accelerate forward? No, the groceries just continued to move forward. That's another example of a virtual force.

Now think of the groves in the record where the coin is sitting. The groves experience a force trying to bend them outward. This is the Centrifugal Force. So if the friction breaks down at 78 RPM and the coin flies off, it would fly in a direction tangential to the curve of the record where it broke loose (until gravity got it to accelerate towards the floor). Not directly outward. When the Centripetal force (friction) breaks down, the Centrifugal would also, and momentum and gravity would be the only things acting on the coin.