Ch. 10 - Rotational Motion

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 10 - Rotational Motion

Problem 14

Chapter 10, Problem 14

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?

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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Related Practice

Textbook Question

The axle of a wheel is mounted on supports that rest on a rotating turntable as shown in Fig. 10–52. The wheel has angular velocity ω₁ = 48.0 rad/s about its axle, and the turntable has angular velocity ω₂ = 35.0 rad/s about a vertical axis. (Note arrows showing these motions in the figure.) What are the directions of $omega sub 1 right arrow$ and $omega sub 2 right arrow$ at the instant shown?

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Textbook Question

A turntable of radius R₁ is turned by a circular rubber roller of radius R₂ in contact with it at their outer edges. What is the ratio of their angular velocities, ω₁/ω₂?

Textbook Question

(II) A child rolls a ball on a level floor 3.1 m to another child. If the ball makes 12.0 revolutions, what is its diameter?

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Textbook Question

The axle of a wheel is mounted on supports that rest on a rotating turntable as shown in Fig. 10–52. The wheel has angular velocity ω₁ = 48.0 rad/s about its axle, and the turntable has angular velocity ω₂ = 35.0 rad/s about a vertical axis. (Note arrows showing these motions in the figure? What is the resultant angular velocity of the wheel, as seen by an outside observer, at the instant shown? Give the magnitude and direction.

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Textbook Question

A child rolls a ball on a level floor 3.1 m to another child. If the ball makes 12.0 revolutions, what is its diameter?

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Textbook Question

The platter of the hard drive of a computer rotates at 7200 rpm (rpm = revolutions per minute = rev/min). If a single bit requires 0.50 μm of length along the direction of motion, how many bits per second can the writing head write when it is 3.00 cm from the axis?

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