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Chapter 8: 8-13P (page 198)

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鈥檚?

Short Answer

Expert verified

The number of revolutions per minute is 33422.5 rpm.

Step by step solution

01

Given data

The distance from the axis of rotation is \({\rm{r}} = 8\;{\rm{cm}}\).

The acceleration is \({\rm{a}} = 100000{\rm{g}}\)

02

 Angular velocity and acceleration

In this problem, to calculate the revolutions per minute of a centrifuge, the relation between radial acceleration and angular velocity will be utilized.

03

Determine the revolutions per minute

The relation to calculate the revolutions per minute is given by:

\(\omega = \sqrt {\frac{a}{r}} \)

Here, r is the radius of the wheel and \(\omega \) is the angular speed.

On plugging the values in the above relation, you get:

\(\begin{aligned}{l}\omega &= \sqrt {\frac{{100000 \times 9.8\;{\rm{m/}}{{\rm{s}}^2}}}{{\left( {8\;{\rm{cm}} \times \frac{{1\;{\rm{m}}}}{{100\;{\rm{cm}}}}} \right)}}} \\\omega &= \left( {3500\;{\rm{rad/s}} \times \frac{{60\;{\rm{rpm}}}}{{2\pi \;{\rm{rad/s}}}}} \right)\\\omega &= 33422.5\;{\rm{rpm}}\end{aligned}\)

Thus, \(\omega = 33422.5\;{\rm{rpm}}\) is the correct answer.

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Most popular questions from this chapter

Two wheels having the same radius and mass rotate at the same angular velocity (Fig. 8鈥�38). One wheel is made with spokes so nearly all the mass is at the rim. The other is a solid disk. How do their rotational kinetic energies compare?

(a) They are nearly the same.

(b) The wheel with spokes has about twice the KE.

(c) The wheel with spokes has higher KE, but not twice as high.

(d) The solid wheel has about twice the KE.

(e) The solid wheel has higher KE, but not twice as high.

FIGURE 8-38

MisConceptual Question 7.

Assume that a 1.00-kg ball is thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, as shown in Fig. 8鈥�46. The ball is accelerated uniformly from rest to 8.5 m/s in 0.38 s, at which point it is released. Calculate (a) the angular acceleration of the arm and (b) the force required for the triceps muscle. Assume that the forearm has a mass of 3.7 kg, and it rotates like a uniform rod about an axis at its end.

FIGURE 8-46

Problems 35 and 36

The platter of the hard drive of a computer rotates at 7200 rpm (rpm = revolutions per minute = rev/min). (a) What is the angular velocity \(\left( {{{{\bf{rad}}} \mathord{\left/{\vphantom {{{\bf{rad}}} {\bf{s}}}} \right.} {\bf{s}}}} \right)\) of the platter? (b) If the reading head of the drive is located 3.00 cm from the rotation axis, what is the linear speed of the point on the platter just below it? (c) If a single bit requires \({\bf{0}}{\bf{.50}}\;{\bf{\mu m}}\) 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?

Question: (II) A 4.2-m-diameter merry-go-round is rotating freely with an angular velocity of \({\bf{0}}{\bf{.80}}\;{{{\bf{rad}}} \mathord{\left/{\vphantom {{{\bf{rad}}} {\bf{s}}}} \right.} {\bf{s}}}\). Its total moment of inertia is \({\bf{1360}}\;{\bf{kg}} \cdot {{\bf{m}}^{\bf{2}}}\). Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. (a) What is the angular velocity of the merry-go-round now? (b) What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?

A cyclist accelerates from rest at a rate of \({\bf{1}}{\bf{.00}}\;{\bf{m/}}{{\bf{s}}^{\bf{2}}}\). How fast will a point at the top of the rim of the tire (diameter = 0.80 cm) be moving after 2.25 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest 鈥� sees Fig. 8鈥�57.]

FIGURE 8-57 Problem 79
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