Question

A centrifuge in a medical laboratory rotates at an angular speed of \(3600 .\) rpm (revolutions per minute). When switched off, it rotates 60.0 times before coming to rest. Find the constant angular acceleration of the centrifuge.

Short answer

Answer: The constant angular acceleration of the centrifuge is -15π rad/s².

Step-by-step solution

Convert the given angular speed to radians per second

The initial angular speed is given in revolutions per minute. To convert it to radians per second, we can use the following conversion factors: - 1 revolution = \(2\pi\) radians - 1 minute = 60 seconds Initial angular speed, \(\omega_0 = 3600 \, \text{rpm}\) Let's convert this to radians per second: \(\omega_0 = 3600 \cdot \frac{2\pi \, \text{radians}}{\text{revolution}} \cdot\frac{1}{60 \, \text{min}}\) \(\omega_0 = 60\pi \, \text{rad/s}\) Now we have the initial angular speed in radians per second.

Determine the distance in radians

The centrifuge rotates 60 times before coming to rest. We can convert this distance to radians by using the conversion factor 1 revolution = \(2\pi\) radians. Distance, \(\theta = 60 \, \text{revolutions} \cdot \frac{2\pi \,\text{radians}}{\text{revolution}}\) \(\theta = 120\pi \,\text{radians}\) Now we have the distance covered in radians.

Use the equations of rotational motion

We can use the following equation of rotational motion to find the angular acceleration (α): \(\omega^2 = \omega_0^2 + 2 \alpha \theta\) The final angular speed (ω) is 0 since the centrifuge comes to rest. We have calculated the initial angular speed (ω₀) and distance (θ) in the previous steps. So we can now solve this equation for the angular acceleration (α). \(0^2 = (60\pi)^2 + 2 \alpha (120\pi)\) \(0 = 3600\pi^2 + 240\pi \alpha\) \(\alpha = -\frac{3600\pi^2}{240\pi}\) \(\alpha = -15\pi \, \text{rad/s}^2\) Hence, the constant angular acceleration of the centrifuge is -15π rad/s² (negative sign indicates that it is a deceleration).

Key concepts

Rotational Motion

Understanding rotational motion is crucial for analyzing the behavior of rotating objects like wheels, turbines, and, as in the exercise at hand, centrifuges. Unlike linear motion, which involves moving in a straight path, rotational motion involves an object spinning about an axis.

Rotational motion can be characterized by variables similar to those used in linear motion. In the given exercise, we talk about angular speed and angular acceleration. Angular speed refers to how fast an object rotates, expressed in radians per second (rad/s) or revolutions per minute (rpm). Rotational motion also obeys its own set of equations similar to the straight-line equations of motion, but with a twist—instead of dealing with distances and velocities, we deal with angles and angular speeds.

Converting Units

When working with rotational motion, it's common to encounter different units of measurement, as seen with the centrifuge's speed initially being presented in revolutions per minute (rpm). Converting these units to a standard form, usually radians per second, is vital because it allows consistent use of the rotational equations of motion.

For angular speed, the conversion process involves understanding that one full revolution equals \(2\text{π}\) radians and that there are 60 seconds in one minute. By multiplying the angular speed by \(2\text{π}/60\), one can convert from rpm to rad/s. Mastery of unit conversion ensures accuracy in physics problems and understanding across various scientific disciplines where different units are prevalent.

Equations of Motion

The equations of motion for rotation describe the relationship between angular displacement, angular velocity, angular acceleration, and time. For the centrifuge exercise, we used a motion equation analogous to one of the linear equations of motion: \(v^2 = u^2 + 2as\), where 'v' is the final velocity, 'u' is the initial velocity, 'a' is the acceleration, and 's' is the displacement.

Translating to Rotational Motion

For rotational motion, this equation translates to \(\omega^2 = \omega_0^2 + 2\alpha\theta\), where \(\omega\) is final angular velocity, \(\omega_0\) is initial angular velocity, \(\alpha\) is angular acceleration, and \(\theta\) is angular displacement. By plugging in the known values and solving for \(\alpha\), we determine the centrifuge's angular acceleration.

Note that the negative angular acceleration indicates deceleration, which makes sense as the centrifuge is coming to rest. The equations of motion are powerful tools in physics, providing predictable outcomes for various objects in motion under constant acceleration.