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Suppose we have a
rigid body, and it's

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rotating about a
fixed axis, and we

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know that the angular
acceleration, alpha, is given.

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Now, it may be a
function of time.

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And we'd like to find
from the angular velocity

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and how much angle has
rotated in some time interval.

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So the first thing
we have to do is

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choose a point in
the rigid body,

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and introduce a
coordinate system.

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Here we have our angle theta.

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And with this point, we have
our r hat and our theta hat

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direction.

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And recall that by
the right-hand rule,

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the k hat is going out of
the plane of the figure.

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And so we have our
coordinate system.

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And we wrote alpha as the
second derivative of theta

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with respect to time k hat.

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Now our goal is to
find omega, which

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we'll right as the first
derivative of d theta dt k hat,

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and also to find theta
as a function of time.

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Now recall that our notation
was that alpha z was the omega

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z dt equals d squared
theta dt squared.

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And in this notation, omega
z was equal to d theta dt.

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Now what we're
going to do is we're

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going to integrate
alpha, just like we

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did in simple circular
motion kinematics

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for point-like objects.

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And so what we have here is
that for some time interval,

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omega z at time t minus
omega z0 is the integral of t

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prime equals 0 to
t of alpha z, which

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is our component of the
angular acceleration.

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We have an integration
variable, dt prime.

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And recall that that
implies that omega z t can

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have some initial
value at time t

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equals 0 plus this
integral relationship,

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which is what we want to figure
out by direct integration.

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Now, this only works when alpha
z is some function of time.

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In order to get the angle,
theta of t, we integrate again,

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where we have theta t0
is the integral from t

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prime equal 0 to t prime equals
t of this function omega z.

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Again, we have some integration
variable, t prime, dt prime.

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And so we see that
theta t can have

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some initial value plus
this integral relationship,

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t prime equals t of
omega z t prime dt prime.

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And this is how
we can figure out

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how the point p has a
component of angular velocity,

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and what angle the
point p sweeps out

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in some time interval t.