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Centripetal acceleration is a key concept when dealing with objects in circular motion. It refers to the acceleration that keeps an object moving along a circular path. This acceleration points towards the center of the circle, a direction known as "radial" or "inward." It is responsible for changing the direction of the object's velocity continuously, resulting in circular motion.
To calculate centripetal acceleration, we use the formula:

• \( a = r\omega^2 \)

where \( a \) is the centripetal acceleration, \( r \) is the radius of the circle (distance from the axis), and \( \omega \) is the angular velocity. This formula shows that centripetal acceleration depends on how fast the object is rotating (angular velocity) and how far it is from the center of rotation (radius).
In our exercise, the acceleration was given in terms of \( g \), where \( g = 9.8 \, m/s^2 \). So, if a centrifuge needs to provide an acceleration of \( 100,000 \, g \), we multiply \( 100,000 \) by \( 9.8 \, m/s^2 \) to find the actual acceleration in \( m/s^2 \). Understanding this process is crucial for solving problems involving rotational motion.

Angular velocity is the measure of how quickly something is spinning or rotating around a central point. When dealing with circular motion, angular velocity indicates the angle through which an object rotates in a certain amount of time. It is usually measured in radians per second (\( rad/s \)).
The connection between angular velocity and centripetal acceleration is pivotal because it helps us find how fast an object must spin to achieve a certain acceleration. Using the formula:

• \( \omega = \sqrt{\frac{a}{r}} \)

we can determine the angular velocity \( \omega \) needed to produce a given centripetal acceleration \( a \) at a radius \( r \).
Given the problem conditions, substituting \( a = 100,000 \times 9.8 \) and \( r = 0.07 \) meters into the formula will provide us with the angular velocity required to give a particle the specified acceleration. This is a crucial step before we can switch from radians per second to revolutions per minute (RPM).

Revolutions per minute (RPM) is a common unit of rotational speed or rate of spin, indicating how many complete turns an object makes in one minute. It is widely used to describe the rotation speed of mechanical components like centrifuges, motors, and wheels.
To convert angular velocity from radians per second to RPM, we use the conversion formula:

• \( \text{RPM} = \frac{\omega}{2\pi} \times 60 \)

where \( \omega \) is the angular velocity in \( rad/s \). This formula considers that one complete revolution is \( 2\pi \) radians and adjusts the time frame from seconds to one minute.
For example, in our solution, we calculated that \( \omega \) was approximately \( 37,412 \; rad/s \). Substituting into the formula, we found the RPM to be about \( 357,900 \; \text{RPM} \). This conversion is essential in practical applications, as RPM is a more intuitive way to express rotational speed, especially in engineering and everyday life contexts.