Acceleration and rotating bodies.

[Niles]

Homework Statement

Ok, when talking about rotating bodies, we deal with the following accelerations - please correct me if I am wrong:

A radial acceleration (a.k.a. the centripetal-acceleration): w^2*r or v^2/r.

An angular acceleration given by dw/dt.

A tangential acceleration given by r * a_angular

Where does linear acceleration come in? If we e.g. look at a uniform circular motion, it has a radial acc., no angular and then no tangential but it has a linear acceleration because it changes direction all the time?

I am quite confused about linear acceleration, and I can't seem to find it described anywhere.

 

[Niles]

Ok, "The total linear acceleration is the vector sum of tangential plus centripetal accelerations".

So in the case above, the total linear acceleration is the centripetal-acceleration, since it is a uniform circular motion.

 

[Shooting Star]

When a body moves along a space curve and v is the speed at a point, the total vector acceleration is (dv/dt)T + (v^2/r)N, where r is the radius of curvature at that point, T is the unit tangent vector and N is the unit vector along the principal normal. In 2-d motion, there is only one normal dirn to the curve. The linear accn you are talking about is dv/dt along the dirn of T, the unit tangent vector. If there is no change in speed, then that component is zero., and you are left with only centripetal accn along the normal.

 

[Niles]

What is the precise difference between angular and tangential acceleration?
Don't they both increase the speed of the body?

When the tangential acc. is zero, the linear acceleration = radial acceleration - but can we still have an angular acceleration at this point? I know that the unit for angular acc. is rad/s^2 and that it increases the angular velocity (omega) - the unit for tangential acc. is m/s^2, but what does this increase (tangential speed I guess, but what is that)?

 

[Shooting Star]

If you mean centripetal acceleration, then it does not change the speed of the body, but the velocity, which means it changes the dirn of movement keeping the speed same.

Angular accn is the change of the angular velo of a body about a point. That may change the speed of the body. It is defined as alpha=dw/dt. It’s also equal to 1/r(dv/dt)=(tangential accn)/r, where r is the radius of curvature at that point.

 

[Niles]

I do mean the tangential acceleration - what does this do?

Is the tangential velocity given by r*omega?

 

[Shooting Star]

The tangential accn changes the speed of the body.

(Velocity is always tangential.) If v is the vector velo of a point wrt an origin O, then v=w X r, where r is the posn vector of the particle. (X denotes cross product.)

!n 2d cicular motion, v=rw.

 

[Niles]

I don't understand how it is possible for an object to have a constant linear speed (angular velocity) and have an angular acceleration? Like a CD.

 

[Shooting Star]

Angular accn is the change of the angular velo of a body about a point. That may change the speed of the body. It is defined as alpha=dw/dt. It’s also equal to 1/r(dv/dt)=(tangential accn)/r, where r is the radius of curvature at that point.

When linear speed is constant, angular accn is zero. Have you read carefully what I've written so far?

 

[Niles]

I got it - thanks :-)