Looking At Basic Rotational Dynamics And Its Applications (Part 1)

 

A simple first approach to rotational dynamics is to examine a rigid disc rotating about its spin axis (z-axis) at an angular velocity equal to w. Let r denote the distance of the axis of symmetry of the disc to any other point in the disc. If a point on the circumference of the rotating disc moves from A to B -  as in the diagram below:

So that the radius OA moves through an angle q  - then its angular velocity w about O is defined as the change of the angle per unit time, e.g.

w =  q/t

If t is in seconds then is usually expressed in radians per second or rad/sec

The period of the motion, T, is given by:  T = 2p /w

Or the time for the point to complete the circle once. (Since  2p = 360 deg by definition)

Let s be the length of the arc AB (see diagram) then s/r =  q by definition then:

 s =  r q .  

Now, we can divide both sides by time t and obtain:

s/ t =  r q / t

But  s/ t =   v  the velocity of the rotating point on the disc and  q /t  is the angular velocity w  so that we can write:  v =  r q / t =  r w.

Note that for a uniform disc such as shown, all the mass elements  (Dm ) will have the same angular velocity   and hence the same angular acceleration, a  .  Note the angular velocity vector is in the same direction as the z-axis, hence is denoted  w _z .

 

A simple practical example of rotation about a fixed axis is the motion of a compact disc in a CD player, which is driven by a motor inside the player. In a simplified model of this motion, the motor produces angular acceleration, causing the disc to spin. As the disc is set in motion, resistive forces oppose the motion until the disc no longer has any angular acceleration, and the disc now spins at a constant angular velocity. Throughout this process, the CD rotates about an axis (z) passing through the center of the disc, and which is perpendicular to the plane of the disc. This type of motion is called fixed-axis rotation.

The angular acceleration occurs if there is a change in angular velocity, i.e.

a =  (w ₁  -  w _o) /t  (in rad/sec² )

A change from an initial  w _o is expressed in terms of the angle q .

q  =  w _o t  +   ½ at²  

Conservation of angular momentum also applies to rotational motion. It states that the total angular momentum of a system is constant if the resultant external torque T_ex acting on the system is zero:

Thus IF: S d(T)_ext/ dt = dL/dt = 0

Then: I_i(w_i) = I_f (w_f )= const.

Where I_i,f are the initial and final moments of inertia, and w_i,_f  denote initial and final angular velocities.  The law (expression) is valid for rotations about a fixed axis or about an axis through the center of mass of the system. 

For a disc of uniform mass  M we have: I =  ½ Mr ²                                                                    

The angular momentum is defined: L = Mw r ²

For a possible change in angular momentum we then need to know I, e.g. for:

I_f (w_f ) -  I_i(w_i) = M w_f r ²   -   M w_i r ²

 To see if:  I_f (w_f ) -  I_i(w_i) =  0

Or:   I_i(w_i) = I_f (w_f )= const.   

Note the kinetic rotational energy:   K  = ½ Iw ²   =   ½M w ² r ²

  is also conserved, including for small mass elements m_i  at distances r _i from the axis of rotation, i.e.

It is not necessary to write a separate angular speed w_i for each element because all mass elements of the uniform disk have the same angular speed, w.

Suggested Problems:

1) An old-fashioned record turntable in the form of a uniform disk of 1.0 kg mass and radius R = 0.1m turns freely with an angular velocity of 8 rad/sec (about 78 rpm).  A dead bumble bee (from bug spray) lands at a point halfway between the center and the outside edge. If the mass of the dead bee is 0.1 kg, find the final angular velocity.

2) A flywheel is speeded up uniformly to 900 rpm (30p rad/sec) in 15 secs from rest. Find the angular acceleration a  in rad/sec² .

3) A disc revolves with a constant acceleration of  5 rad/sec² .  How many turns does it make in 8 seconds from rest?

4) A compact disc has an angular speed of 210 rpm. Find it angular velocity in rad/s.

5) A compact disc (CD) speeds up uniformly from rest to 310 rpm in 3.3 s.  How would you calculate the number of revolutions the CD makes in this time? How many would that be?

6) A massive uniform disk (e.g. grindstone) of radius R = 0.610 m has a moment of inertia I = 4.3 kg · m².  Find the mass of the grindstone.

7) A flywheel of mass 1.0 kg and radius 0.1m experiences a sudden change in angular velocity (from rest, i.e. w_i = 0 rad/s)  to  w_f = 20 rad/sec.  Has an external torque been applied to cause this?