Suppose we have a rigid body, and it's
rotating about a fixed axis, and we
know that the angular acceleration, alpha, is given.
Now, it may be a function of time.
And we'd like to find from the angular velocity
and how much angle has rotated in some time interval.
So the first thing we have to do is
choose a point in the rigid body,
and introduce a coordinate system.
Here we have our angle theta.
And with this point, we have our r hat and our theta hat
direction.
And recall that by the right-hand rule,
the k hat is going out of the plane of the figure.
And so we have our coordinate system.
And we wrote alpha as the second derivative of theta
with respect to time k hat.
Now our goal is to find omega, which
we'll right as the first derivative of d theta dt k hat,
and also to find theta as a function of time.
Now recall that our notation was that alpha z was the omega
z dt equals d squared theta dt squared.
And in this notation, omega z was equal to d theta dt.
Now what we're going to do is we're
going to integrate alpha, just like we
did in simple circular motion kinematics
for point-like objects.
And so what we have here is that for some time interval,
omega z at time t minus omega z0 is the integral of t
prime equals 0 to t of alpha z, which
is our component of the angular acceleration.
We have an integration variable, dt prime.
And recall that that implies that omega z t can
have some initial value at time t
equals 0 plus this integral relationship,
which is what we want to figure out by direct integration.
Now, this only works when alpha z is some function of time.
In order to get the angle, theta of t, we integrate again,
where we have theta t0 is the integral from t
prime equal 0 to t prime equals t of this function omega z.
Again, we have some integration variable, t prime, dt prime.
And so we see that theta t can have
some initial value plus this integral relationship,
t prime equals t of omega z t prime dt prime.
And this is how we can figure out
how the point p has a component of angular velocity,
and what angle the point p sweeps out
in some time interval t.