Question

A centrifuge has a rotational inertia of $6.5 \times 10^{-3} \mathrm{kg} \cdot \mathrm{m}^{2}$ How much energy must be supplied to bring it from rest to 420 rad/s \((4000 \text { rpm }) ?\)

Short answer

\) Answer: The energy required to bring the centrifuge from rest to an angular speed of \(420\,\mathrm{rad/s}\) is approximately \(572.94\,\mathrm{J}.\)

Step-by-step solution

Understand the important parameters and energy conservation principle

The exercise provides us with the rotational inertia (I) of the centrifuge, \(6.5\times10^{-3}\,\mathrm{kg}\cdot\mathrm{m}^2,\) and the final angular speed (ω), \(420\,\mathrm{rad/s}.\) The centrifuge is initially at rest, which means its initial angular speed (\(\omega_0\)) is \(0\,\mathrm{rad/s}.\) Using the conservation of energy, we can calculate the energy needed to bring the centrifuge from rest to the given angular speed.

Calculate the initial and final rotational energies

First, we will calculate the initial and final rotational energies of the centrifuge using the formula for rotational kinetic energy: Rotational Kinetic Energy, \(K=\frac{1}{2}I\omega^2.\) Initial Rotational Energy (\(K_0\)) = \(\frac{1}{2}I\omega_0^2 = \frac{1}{2}\times(6.5\times10^{-3}\,\mathrm{kg}\cdot\mathrm{m}^2)\times0^2=0\,\mathrm{J}\) Final Rotational Energy (\(K_f\)) = \(\frac{1}{2}I\omega^2 = \frac{1}{2}\times(6.5\times10^{-3}\,\mathrm{kg}\cdot\mathrm{m}^2)\times(420\,\mathrm{rad/s})^2\)

Calculate the energy that must be supplied to the centrifuge

Now we will calculate the required energy to bring the centrifuge from rest to the given angular speed using the conservation of energy principle: Energy supplied (\(E\)) = Final Rotational Energy (\(K_f\)) - Initial Rotational Energy (\(K_0\)) \(E= K_f - K_0\) Substitute the values of \(K_0\) and \(K_f\) into the equation and compute the energy supplied: \(E = \frac{1}{2}\times(6.5\times10^{-3}\,\mathrm{kg}\cdot\mathrm{m}^2)\times(420\,\mathrm{rad/s})^2 - 0\,\mathrm{J}\) \(E \approx 572.94\,\mathrm{J}\) So, the energy that must be supplied to bring the centrifuge from rest to 420 rad/s is approximately \(572.94\,\mathrm{J}.\)