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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

One of the problems posed by the Marquis de I’Hospital in his calculus textbook Analyse des Infiniment Petits concerns a pulley that is attached to the ceiling of a room at a point C by a rope of length r. At another point B on the ceiling, at a distance d from C (where d > r), a rope of length is attached and passed through the pulley at F and connected to a weight W. The weight is released and comes to rest at its equilibrium position D. (See the figure.) As I’Hospital argued, this happens when the distance | ED | is maximized. Show that when the system reaches equilibrium, the value of x is

r 4 d ( r + r 2 + 8 d 2 )

Chapter 4, Problem 23P, One of the problems posed by the Marquis de IHospital in his calculus textbook Analyse des

To determine

To show: For Marquis de L’Hospital’s model when the system reaches equilibrium the value of x is r4d(r+r2+8d2) .

Explanation

Given:

Figure showing the arrangements of Marquis de L’Hospital’s model.

Let DB be the rope of length l and length of EF be a and BF be b.

Thus observe that,

l=BD=BF+FD

Similarly calculate as follows:

ED=EF+FD=a+lb=r2x2+l(dx)2+a2=r2x2+l(dx)2+(r2x2)2

Simplify:

ED=r2x2+l(dx)2+(r2x2)2=r2x2+ld22dx+x2+(r2x2)=r2x2+ld22dx+r2

Differentiate ED with respect to x.

dEDdx=2x2r2x2+2d2d22dx+r2=dd22dx+r2xr2x2

Set dEDdx=0 and obtain that,

dd22dx+r2xr2x2=0dd22dx+r</

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