Rotational Motion

We begin by studying rotational kinematics. Remember kinematics - the study of how velocities and accelerations are related in the motion of a system - we studied linear kinematics earlier.  Hello??? And dynamics is the study of how those motions arise.

Rotational motion is an analog to linear motion: in linear motion we had displacement, velocity, acceleration, etc. and in rotational motion we also have displacement, velocity, and acceleration BUT they’re called angular displacement, angular velocity, and angular acceleration.

Let’s see how these terms are defined. We start, appropriately, with a circle:

We notice that if this disk rotates, like a CD or a floppy disk does, points 1a and 2a must turn together to become points 1b and 2b, respectively. Notice that point 1, though, has to travel a greater distance than point 2. Yet it must do so in the same time as point 2. Obviously it must be moving at a higher velocity...and in fact it is!! That means different points along the radius travel at different velocities!!!  But also note that the angle q is the same for both points 1 and 2...So if we use the number of radians per second to denote velocity, then we find all points on the rotating disk move at the same speed (angular speed).

So, here's what we do: instead of Dx as displacement, we use Dq as angular displacement (in radians).

Recall that linear velocity was: v = Dx/Dt 

(displacement/time - sometimes referred to as "rate of change of displacement")

Now we define angular velocity as: w = Dq/Dt    (w is Greek lower-case "omega") 

(angular displacement/time, or "rate of change of angular displacement")

Recall that linear acceleration was: a = Dv/Dt 

(change in velocity/time, or, yes again, "rate of change of velocity")

And we define angular acceleration as: a = Dw/Dt (a is Greek "alpha")

(rate of change of angular velocity)

Eventually we’re going to define analogies for all the linear quantities, such as inertia, force, momentum, etc. A table summarizing these is shown below, but be patient: it will take time to explain where those other quantities come from:

where the unidentified quantities in the table are,

r = radius

m = mass

For now, we will use the first three quantities to give us the kinematic equations in rotation:

The form of the rotational equations is exactly the same as the linear ones we studied earlier...and the handling is just the same. You only need to remember that w must be in radians per second (r/s) and a in r/s².

Example:

A metal cylinder of radius 0.45 meters is spinning at 2000 rpm and a brake is applied slowing it to 1000 rpm in 10 seconds. A)what is the angular acceleration? B)how many revolutions does it make before it reaches 1000 rpm? C)how far does a point on the edge of the disk travel during this time?

Answer:

The first thing to do is convert 2000 rpm (revolutions per minute) and 1000 rpm into 

radians per second:

In one revolution there are 2p radians (or 360 degrees)

In one minute there are sixty seconds

So,  (2000 rev/min) x (2p rad/sec) x (1 min/60 sec) = 209.4 r/sec

And (1000 rev/min) x (2p rad/sec) x (1 min/60 sec) = 104.7 r/sec

Now we can use the equations we mentioned…

a) w_f = w_i + at

(104.7 r/s) = (209.4 r/s) + a(10 s)

a = 10.5 r/s²

b) Dq = w_it + (1/2)at²

= (209.4 r/s)(10 s) + (1/2)(10.5 r/s²)(10 s)²

= 2619 radians

We must convert this to the number of rotations! (How do we do that?)

...Each rotation contains 2p radians, so 2619/(2p) = 417 rotations

c) Since we're talking about distance in meters then we need to calculate how many meters we move in one rotation...then we can find out how many meters are in 417 rotations:

Meters in one rotation = 2pr (circumference) = 2.83 meters

So the total distance traveled = (417 rotations)(2.83 meters/rotation)

= 1179 meters.

(this could also be used to determine how far a car moves if it's tires rotate a certain number of times - it translates directly into linear motion)