You have been hired as part of a research team consisting of biologists, compute
ID: 2243465 • Letter: Y
Question
You have been hired as part of a research team consisting of biologists, computer scientists, engineers, mathematicians, and physicists investigating the virus which causes AIDS. This effort depends on the design of a new centrifuge which separates infected cells from healthy cells by spinning a container of these cells at very high speeds. Your design team has been assigned the task of specifying the mechanical structure of the centrifuge arm which holds the sample container. For aerodynamic stability, the arm must have uniform dimensions. Your team decided the shape will be a long, thin strip of length L, width w, and thickness t. The mass of the strip is M. The actual values of these quantities will be optimized by a computer program. For mechanical reasons, the arm must be stronger at one end than at the other. Your team decided to use new composite materials to accomplish this. Using these materials changes the strength by changing the density of the arm along its length while keeping its dimensions constant. To calculate the strength of the brackets necessary to support the arm, you must determine the position of the center of mass of the arm. You decide to do this in two different ways.
a- do calculation by assuming that arm is a continuous material with a density which varies linearly along its length as (A + Bx).
Explanation / Answer
The length of the strip = L i.e, x changes from 0 to L.
Taking a small elment of width dx, at a distance of x from the point x = 0.
Mass of this small element , dm = linear density * length = (A + Bx) * dx
= Ax + [Bx^2 / 2].
Integrating from 0 to L, we get total mass as M = AL + BL^2 / 2.
Now, centre of mass = integral of ( x.dm ) / M
= [ AL^2 / 2 + BL^3 / 6 ] / [ AL + BL^2 / 2 ]
Taking L common, we get CM = [3AL + BL^2 ] / [6A + 3BL ].
THIS IS THE CENTRE OF MASS ACCORDING TO THE SECOND PART.
By the first part,
Centre of mass for discrete bodies = sum of [m.x / M]
Thus, CM = [ m.0 + m.L/2 + m.3L/4 + m.L ] / 4m
= ( 9mL / 4 ) / 4m
= 9L / 16.
Thus, the CM would be at a distance of 9L / 16 distance from the starting end.
THANK YOU. DO RATE.
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