Question

A centrifuge rotor has a moment of inertia of 3.20 10-2 kg·m2. How much energy is required to bring it from rest to 5000 rpm?

Please include formulas

Answer 1

Concepts and reason

The concept required to solve the given problem is work-energy theorem in rotational kinematics.

Initially, convert the angular speed of centrifuge from rpm to radians per second by using conversion factor. Then, use work-energy theorem to find the work done on the centrifuge.

Fundamentals

The rotational kinetic energy K of a rotating object is given by following expression.

$K = \frac{1}{2}I{\omega ^2}$

Here, I is the moment of inertia, and $\omega$is the angular speed.

According to work-energy theorem in rotational kinematics, the work done W on a rotating object is equal to change in rotational kinetic energy of the object.

$W = \frac{1}{2}I\omega _{\rm{f}}^2 - \frac{1}{2}I\omega _{\rm{i}}^2$

Here, I is the moment of inertia, ${\omega _{\rm{f}}}$is the final angular speed, and ${\omega _{\rm{i}}}$is the initial angular speed.

The final angular speed of the centrifuge is,

${\omega _{\rm{f}}} = 5000\,{\rm{rpm}}$

Convert the units for final angular speed from rpm to radians per second.

$\begin{array}{c}\\{\omega _{\rm{f}}} = 5000\,{\rm{rpm}}\left( {\frac{{2\pi \,{\rm{rad}}}}{{1\,{\rm{rev}}}}} \right)\left( {\frac{{1\,{\rm{min}}}}{{60\,{\rm{s}}}}} \right)\\\\ = 523.59\,{\rm{rad/s}}\\\end{array}$

According to work-energy theorem in rotational kinematics, the work done W on a rotating object is equal to change in rotational kinetic energy of the object.

$W = \frac{1}{2}I\omega _{\rm{f}}^2 - \frac{1}{2}I\omega _{\rm{i}}^2$

Substitute 523.59 rad/s for ${\omega _{\rm{f}}},$0 rad/s for ${\omega _{\rm{i}}},$and $3.20 \times {10^{ - 2}}\,{\rm{kg}} \cdot {{\rm{m}}^2}$for I in the above equation.

$\begin{array}{c}\\W = \frac{1}{2}\left( {3.20 \times {{10}^{ - 2}}\,{\rm{kg}} \cdot {{\rm{m}}^2}} \right){\left( {523.59\,{\rm{rad/s}}} \right)^2} - \frac{1}{2}\left( {3.20 \times {{10}^{ - 2}}\,{\rm{kg}} \cdot {{\rm{m}}^2}} \right){\left( 0 \right)^2}\\\\ = 4386.3\,{\rm{J}}\\\end{array}$

Ans:

The work done is 4386.3 J.