The revolutions per minute are\({\omega _2} = 15000\,{\rm{rpm}}\).

The time period is \(t = 240\;{\rm{s}}\).

In this problem, the angular displacement is determined by using the equation of kinematics for the rotational motion of the object. Also, the initial angular speed of the centrifuge is zero because it is at rest.

The relation to find the angular displacement is given by:

\(\theta = \frac{1}{2}\left( {{\omega _1} + {\omega _2}} \right)t\)

Here, \({\omega _1}\) is the initial angular speed, whose value is zero.

On plugging the values in the above relation, you get:

\(\begin{aligned}{l}\theta &= \frac{1}{2}\left( {0 + 15000\,{\rm{rpm}}} \right)\left( {240\;{\rm{s}} \times \frac{{1\,{\rm{min}}}}{{60\;{\rm{s}}}}} \right)\\\theta &= 3 \times {10^4}\;{\rm{rev}}\end{aligned}\)

Thus, \(\theta = 3 \times {10^4}\;{\rm{rev}}\) is the required angular displacement.