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 -60π² radians/second².

Step-by-step solution

Convert initial angular speed to radians per second

We are given an initial angular speed of 3600 rpm (revolutions per minute). To convert this to radians per second, we use the conversion factors for minutes to seconds and revolutions to radians: \[ 1~minute = 60~seconds \] \[ 1~revolution = 2\pi~radians\] So, the conversion is: \[ \omega_0 = 3600\cdot\frac{2\pi~radians}{1~revolution} \cdot \frac{1~minute}{60~seconds} = 2\pi \times 60~radians/second \]

Convert the number of rotations to radians

We are given that the centrifuge rotates 60 times before coming to rest. To convert this into radians, we can use the conversion factor of 1 revolution = 2π radians: \[ \theta = 60\cdot\frac{2\pi~radians}{1~revolution} = 120\pi~radians\]

Use the angular kinematic equation to find the angular acceleration

Since the centrifuge comes to rest, we know that its final angular speed (ω) is 0 radians/second. We can use the following angular kinematic equation to find the constant angular acceleration (α): \[ \omega^2 = \omega_0^2 + 2\alpha\theta \] Rearrange the equation to solve for α: \[ \alpha = \frac{\omega^2 - \omega_0^2}{2\theta}\] Now plug in the values for ω, ω₀, and θ: \[ \alpha = \frac{(0)^2 - (2\pi \times 60)^2}{2(120\pi)} =-\frac{(2\pi \times 60)^2 }{2(120\pi)} \] \[ \alpha = - \frac{4\pi^2 \times 3600}{240\pi} = -60\pi^2 ~radians/second^2 \]

Present the final answer

The constant angular acceleration of the centrifuge is -60π² radians/second².

Key concepts

Kinematic Equations

Kinematic equations are powerful tools in physics that describe the motion of objects. In the realm of angular motion, these equations get adapted to account for rotational motion, linking variables like angular displacement (\theta), angular velocity (\text{\(\omega\)}), and angular acceleration (\text{\(\alpha\)}). It's crucial to understand that angular kinematic equations are very similar to their linear counterparts, but they specifically deal with rotational movement around a circle.

For instance, when the textbook problem mentions that 'it rotates 60.0 times before coming to rest,' what we are really discussing is angular displacement, and an equation of motion links this displacement to initial angular velocity and angular acceleration, allowing us to solve for the unknown variable.

Angular Speed

Angular speed (often represented by the symbol \text{\(\omega\)}) is a measure of how quickly an object rotates or revolves relative to another point, expressed in terms such as revolutions per minute (rpm) or radians per second (rad/s). It can be thought of as the angular equivalent of linear speed.

In our exercise, the centrifuge has an initial angular speed of 3600 rpm. To make this information useful in physics calculations, it must first be converted to the standard unit of radians per second. This is because all kinematic equations require consistency in units to yield correct solutions.

Unit Conversion

Unit conversion is a fundamental skill in physics problem-solving because equations require inputs to be in consistent units. For angular measurements, radians are the standard unit, whereas in everyday scenarios, we often encounter revolutions or turns.

So, when we're given values like 3600 rpm for the angular speed, it's imperative to convert to radians per second as we did by using the conversion factors (\text{\(1~minute = 60~seconds\)}) and (\text{\(1~revolution = 2\pi~radians\)}). Careful conversion ensures that we achieve the accurate angular acceleration. Omitting or incorrectly converting units can lead to errors and incorrect results.

Physics Problem Solving

Solving physics problems involves a systematic approach of understanding concepts, applying appropriate formulas, and performing accurate calculations. Step-by-step reasoning helps break down complex problems into manageable parts, as we saw in the exercise involving a centrifuge's angular acceleration.

1. Interpret the problem and identify knowns and unknowns.
2. Choose a relevant physics principle or law.
3. Apply the appropriate formula or equation.
4. Perform precise unit conversions.
5. Insert known values and solve for the unknown.
6. Evaluate your solution to ensure it makes sense.

By applying this structured approach, we tackled the challenge of finding the angular acceleration of a decelerating centrifuge, showcasing how each step builds towards the final solution.