Problem 22 - College Physics

Question

Find the required angular speed (in rpm) of an ultracentrifuge for the radial acceleration of a point \(2.50 \mathrm{~cm}\) from the axis to equal \(400,000 \mathrm{~g}\)

Step-by-Step Solution

Answer

The angular speed required is approximately 47,381 rpm.

Step 1: Convert Acceleration from g to m/s²

We know that acceleration due to gravity, \( g \), is approximately \( 9.8 \text{ m/s}^2 \). Therefore, \( 400,000 \ g \) is \( 400,000 \times 9.8 \text{ m/s}^2 \). Calculate this value to convert the acceleration to m/s².

Step 2: Convert Distance to Meters

The distance from the axis is given in centimeters and needs to be in meters for standard SI units. So, convert \( 2.50 \text{ cm} \) to meters: \( 2.50 \text{ cm} = 0.025 \text{ m} \).

Step 3: Use Radial Acceleration Formula to Find Angular Velocity

The radial acceleration \( a \) is given by \( a = \omega^2 r \), where \( \omega \) is the angular velocity in radians per second, and \( r \) is the radius in meters. Rearrange the formula to solve for \( \omega \): \( \omega = \sqrt{\frac{a}{r}} \). Substitute \( a = 400,000 \times 9.8 \) m/s² and \( r = 0.025 \) m into the equation to find \( \omega \).

Step 4: Convert Angular Velocity to Revolutions Per Minute (rpm)

First, convert the angular velocity from radians per second to revolutions per second by dividing by \(2\pi\), since there are \(2\pi\) radians in a revolution. Then, convert this to revolutions per minute by multiplying by 60 seconds. This gives the angular speed in rpm.

Key Concepts

Centrifugal Force

Centrifugal force is often encountered when objects move in a circle. Imagine spinning around quickly while holding onto a rope attached to an object. The tendency of the object is to fly outwards away from the circle. This apparent force that pushes the object outward is what we call centrifugal force. It is often considered a "fictitious" force because it arises from the object's inertia and the circular path you are forcing it to follow.

Essentially, when an object is moving in a circular path, its inertia wants it to continue in a straight line. However, a continuous force (often called the centripetal force) keeps it in the circle. The centrifugal force is the perception of the object's desire to move straight, particularly felt by an observer within the rotating system.

• Occurs in rotating systems.
• Fictitious force due to inertia.
• Always directed away from the axis of rotation.

Understanding centrifugal force is crucial when studying rotating machinery like ultracentrifuges or even when designing rides in amusement parks.

Radial Acceleration

Radial acceleration, also known as centripetal acceleration, is the acceleration that points towards the center of a circular path. It is crucial in maintaining an object's circular motion by continually changing its velocity direction.

Radial acceleration is quantified by the formula \[ a = \omega^2 r \]where - \( a \) is the radial acceleration,- \( \omega \) is the angular velocity in radians per second,- \( r \) is the radius from the center of the circle to the point object.
Given that this concept is integral to understanding circular motion, it explains why an object in a circular path needs constant force applied towards the center to keep moving in a circle. This force is necessary because the velocity direction is always changing, even if the speed is constant.

Angular Speed Calculation

Angular speed is a measure of how quickly an object is rotating. It is typically represented in radians per second or revolutions per minute (rpm). When solving problems related to rotation, angular speed is key.

To calculate angular speed, we can use radial acceleration as follows: Rearrange the radial acceleration formula,\[ a = \omega^2 r \]to find\[ \omega = \sqrt{\frac{a}{r}} \],where \( \omega \) is the angular speed and \( r \) is the radius.

• Measure angular speed in radians/second or rpm.
• Useful to know in systems with mechanical rotation like engines or centrifuges.

Understanding the calculation of angular speed helps in designing and operating rotating systems and ensures they function within safe limits.

SI Units Conversion

Converting measurements into the International System of Units (SI) forms the backbone of scientific calculations. Consistency in units ensures calculations are correct and comparable.

For example, when dealing with distances, it's crucial to convert units to meters, the standard SI unit for length. If you have a measurement in centimeters, you convert it by dividing by 100:\[ \text{{Centimeters}} \to \text{{Meters}} \]Similarly, accelerations given in terms of \( g \) (where \( g \approx 9.8 \text{ m/s}^2 \)) should be converted to \( \text{m/s}^2 \).

• Standardize units to meters, kilograms, seconds.
• Facilitates accurate and consistent calculations.
• Aids in understanding and comparing scientific results.

Through these conversions, calculations become universally understandable and operable within the scientific community.