Problem 25 - University Physics with Modern Physics

Question

An advertisement claims that a centrifuge takes up only \(0.127 \mathrm{~m}\) of bench space but can produce a radial acceleration of \(3000 \mathrm{~g}\) at 5000 rev \(/ \mathrm{min}\). Calculate the required radius of the centrifuge. Is the claim realistic?

Step-by-Step Solution

Answer

Calculate the radius as approximately 0.486 m and find that the claim of 0.127 m is unrealistic.

Step 1: Identify Known Values

We know the radial acceleration as a multiple of Earth's gravity, given as \(3000 \mathrm{~g}\). Each \(\mathrm{g}\) is \(9.81 \, \mathrm{m/s^2}\), so the radial acceleration is \(3000 \times 9.81 \, \mathrm{m/s^2}\). The centrifuge rotates at \(5000 \mathrm{~rev/min}\).

Step 2: Convert Rotation Speed

Convert the given rotation speed from revolutions per minute to radians per second. One revolution is \(2\pi\) radians and there are 60 seconds per minute. So, the angular velocity \(\omega\) is \(\frac{5000 \times 2\pi}{60} \, \mathrm{rad/s}\).

Step 3: Relation Between Acceleration and Radius

Use the formula for radial acceleration \(a = \omega^2 r\), where \(r\) is the radius and \(a\) is the radial acceleration. Substitute the values to get \(3000 \times 9.81 = \left(\frac{5000 \times 2\pi}{60}\right)^2 r\).

Step 4: Solve for Radius

Rearrange the formula to find \(r\). Solve for \(r\) by dividing both sides by the angular velocity squared. This gives us \(r = \frac{3000 \times 9.81}{\left(\frac{5000 \times 2\pi}{60}\right)^2}\).

Step 5: Calculate the Radius

Calculate \(r\) with the values inserted, ensuring units are consistent. This will give the required radius for such a radial acceleration at the specified rotational speed.

Step 6: Evaluate the Claim

Compare the calculated radius to the claimed bench space of 0.127 m. If \(r\) is significantly less than or equal to half of 0.127 m, the claim is realistic, otherwise it is not.

Key Concepts

Radial Acceleration Calculation

Radial acceleration is the acceleration experienced by an object moving in a circular path, directed towards the center of rotation. In the centrifuge context, it's a measure of how fast something moves in a circle. Acceleration is expressed in terms of Earth's gravity (where 1g is approximately 9.81 m/s²).
To find the radial acceleration from the given centrifugal force, we first identify the known values. Here, a radial acceleration of 3000g is given. This translates to:

• Radial acceleration = 3000 imes 9.81 \, \mathrm{m/s^2}

Using the formula for radial acceleration \( a = \omega^2 \times r \), we solve for the radius \( r \), after finding \( \omega \) from the angular velocity conversion. This equation helps connect physical rotation measures to linear acceleration in circular paths.

Angular Velocity Conversion

Converting the given rotation speed of the centrifuge is crucial for accurate calculations. The speed is initially given in revolutions per minute (rpm), which isn't directly usable in physics equations involving radians. To convert:

• 1 revolution = 2π radians
• 1 minute = 60 seconds

Given 5000 revolutions per minute, the angular velocity \( \omega \) becomes:

• \( \omega = \frac{5000 \times 2\pi}{60} \, \mathrm{rad/s} \)

This conversion helps us to get the angular velocity in radians per second, a unit that's harmonized with other standard SI units like meters and seconds. Without this conversion, calculating accurate radial acceleration isn't possible.

Rotational Motion

Rotational motion involves any object moving in a circular path. For objects like centrifuges, understanding rotational dynamics is key to comprehending forces at play. Centrifuges create strong radial accelerations to separate substances, such as blood components with different densities, by spinning rapidly.
In physics, rotational motion is characterized by parameters like:

• Angular velocity: How fast an object spins
• Radial acceleration: The inwards force experienced by the rotating object
• Radius of rotation: Distance from the center of the circle to the path of motion

Understanding these aspects allows you to solve equations concerning rotating objects and assess the realism of various claims about such systems, like evaluating if a centrifuge's required radius fits into the advertised bench space. The interplay between these factors determines the outcomes for devices operating under rotational motion.