Rotation About a Fixed Axis

An example of a rigid body that is rotating about a fixed axis could be a car wheel. A rigid body that is rotating around a fixed axis has no motion at the axis that it is rotating around. However, as you look at particles that are a certain distance away from the axis they have an angular motion, and as this distance increases their tangential and normal components will increase the further the particle is from the axis of rotation.

Before, I show how to calculate the tangential or normal components of a specific particle; I would like tod discuss the angular components, since they are used to calculate the tangential components and normal components. The first thing that you must realize is all angular components must be in radians. To change degrees over to radians equation 1 would be used.

(Eq 1)  $degrees = radians\left(\frac{180^o}{π}\right)$

$1~rev=360^o=2π$

That being said, your angular displacement would be certain amount of radians from point a to point b.

(Eq 2)  $ for \frac{1}{2}~rev,~θ=π~rad$

The angular velocity would be based off of the time it takes to get from point a to point b. This means angular velocity is independent from the distance to the fixed axis, but instead is directly related to the rpm of the rigid body’s rotation at any given time. To calculate angular velocity equation 3 would be used.

(Eq 3)  $ϖ=\frac{dθ}{dt},~units~\left(\frac{rad}{s}\right)$

Finally, angular acceleration is calculated based of the time it takes for an initial angular velocity to change to a new angular velocity. Again this is independent from the distance to the fixed axis. To calculate angular acceleration refer to equation 4.

(Eq 4) $α=\frac{dϖ}{dt}=\frac{d^2θ}{dt^2},~units~\left(\frac{rad}{s^2}\right)$

Now I mentioned earlier that the angular motion of a particle would relate directly to the tangential components of the particle when it is a certain distance r from the fixed axis. To calculate tangential velocity, and the tangential acceleration equations 5 and 6 would be used.

(Eq 5)  $V_t=ϖ^2r$

(Eq 6) $a_t=ar$

Also, remember from kinematics of a particle, particles that follow a curved path also have an acceleration called normal acceleration. To calculate the normal acceleration of a particle, equation 7 would be used.

(Eq 7) $a_n=ϖ^2r$

Once the normal and tangential acceleration have been determined, the acceleration of the particle can be determined using equation 8.

(Eq 8)  $a=\sqrt{a_n^2+a_t^2}$

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