How fast (in rpm) must a centrifuge rotate if a particle 8.0 cm from the axis of rotation is to experience an acceleration of 100,000 g’s?

검증된 단계별 안내

Understand the problem: We are tasked with finding the rotational speed of a centrifuge in revolutions per minute (rpm) such that a particle located 8.0 cm (0.08 m) from the axis of rotation experiences a centripetal acceleration of 100,000 g's, where g = 9.8 m/s².

Express the centripetal acceleration formula: The centripetal acceleration \( a_c \) is given by \( a_c = r \omega^2 \), where \( r \) is the radius (distance from the axis of rotation) and \( \omega \) is the angular velocity in radians per second.

Rearrange the formula to solve for \( \omega \): \( \omega = \sqrt{\frac{a_c}{r}} \). Substitute \( a_c = 100,000 \times 9.8 \ \text{m/s}^2 \) and \( r = 0.08 \ \text{m} \) into the equation.

Convert \( \omega \) (angular velocity) to revolutions per minute (rpm): First, note that \( \omega \) is in radians per second. To convert to revolutions per second, use \( \text{revolutions per second} = \frac{\omega}{2\pi} \). Then, multiply by 60 to convert to rpm: \( \text{rpm} = \frac{\omega}{2\pi} \times 60 \).

Substitute the calculated \( \omega \) into the rpm formula to find the final rotational speed. Ensure all units are consistent throughout the calculation.

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주요 개념

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Centripetal Acceleration

Centripetal acceleration is the acceleration experienced by an object moving in a circular path, directed towards the center of the circle. It is calculated using the formula a = v²/r, where 'a' is the centripetal acceleration, 'v' is the tangential velocity, and 'r' is the radius of the circular path. In this context, understanding how centripetal acceleration relates to the speed of the centrifuge is crucial for solving the problem.

Gravitational Acceleration

Gravitational acceleration, denoted as 'g', is the acceleration due to Earth's gravity, approximately 9.81 m/s². In this question, the particle experiences an acceleration of 100,000 g's, which means it is subjected to an acceleration of 100,000 times that of gravity. This concept is essential for converting the acceleration into a usable value for calculations involving the centrifuge's speed.

Revolutions Per Minute (RPM)

Revolutions per minute (RPM) is a unit of rotational speed that indicates how many complete turns an object makes in one minute. To find the required RPM for the centrifuge, one must relate the tangential velocity to the angular velocity, which is expressed in RPM. Understanding this relationship is key to determining how fast the centrifuge must rotate to achieve the desired acceleration.