The rotor of the Beams ultracentrifuge (see Problem 53 ) is 20.0 cm long. For a point at the end

The rotor of the Beams ultracentrifuge (see Problem 53 ) is...

The rotor of the Beams ultracentrifuge (see Problem 53 ) is $20.0 \mathrm{cm}$ long. For a point at the end of the rotor, find the (a) initial speed, (b) tangential acceleration component, and (c) maximum radial acceleration component.

Key Concepts

Relationship Between Linear and Angular Speed

This concept explains that the linear speed of any point on a rotating object is directly determined by its angular speed and its distance from the axis of rotation, as given by the relationship v = ? × r. Understanding this relationship is essential in converting between rotational motion (angular speed) and the linear motion experienced at a point on the object.

Tangential Acceleration

Tangential acceleration is the component of acceleration that is tangent to the path of the rotating object and is caused by the change in the angular speed over time. It is quantified by the equation a_t = ? × r, linking the angular acceleration of the object to the linear acceleration at a point located some distance from the axis.

Centripetal (Radial) Acceleration

Centripetal acceleration refers to the acceleration that is directed toward the center of the circular path, necessary for keeping an object moving in a curved trajectory. It is determined by the formula a_c = ?² × r, illustrating how both the square of the angular speed and the radial distance influence the inward acceleration needed to maintain the circular motion.

Rotational Kinematics

Rotational kinematics is the study of angular motion and includes the analysis of quantities such as angular displacement, angular velocity, and angular acceleration. These kinematic relationships provide the framework for analyzing how rotational motion evolves over time, analogous to the linear kinematics of translational motion.

Transcript

00:01 So in this example, we're going to be considering a rotor in a centrifuge where the rotor is sped up to 5 times 10 to the 5th radiance per second.

00:13 And then we shut it off and we just let it slow down over time.

00:18 So it slows down very, very slowly at a rate of minus 0 .4 radiance per second squared.

00:24 And what we want to figure out is initially what is the tangential velocity out at the end of the rotor.

00:32 We want to determine what the linear or tangential acceleration is out there.

00:39 And then we also want to determine what the maximum radial acceleration is.

00:44 So for the first part, finding the initial linear velocity, so we'll call this v .i, all we have to do is just recall that for uniform circular motion, the linear speed at a point on this rotor will be related to the angular speed by v equals r omega.

01:08 In this case, omega initial, because we want to know about the linear speed at the start.

01:17 So if we plug in these values here, we have r is 20 centimeters or 0 .2 meters for the radius and omega.

01:26 Initial is 5 times 10 to the fifth radiance per second.

01:35 We multiply those out.

01:37 We end up with 1 times 10 to the 5th meters per second.

01:44 1 times 10 to the 5th meters per second.

01:53 All right.

01:54 So now similarly, we are going to try to find the tangential or linear acceleration.

02:05 So for this one, the easiest thing to do is just to recall that the tangential acceleration and the angular acceleration are related in a similar way that linear velocity and angular velocity are.

02:21 We have that the tangential acceleration here is going to be equal to r times the angular acceleration...

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