(a) Consider the interval r e (a, bl as a rod of variable density p(x). Let-c be

Question

(a) Consider the interval r e (a, bl as a rod of variable density p(x). Let-c be any point in [a, b] and imagine the entire x-axis as a lever, with fulcrum c. The part of the rod to the right of c exerts force on the lever. Explain why this force is approximated by i-1 as N -oo. Write a definite integral for this force. (b) Likewise, write a definite integral for the force applied to the lever form the part of the rod from r- a to z = c. Write a single definite integral for the total force applied to the lever. (c) Under what conditions will the rod balance perfectly at r- c? The point c is called the center of the mass of the rod. (d) Find the center of the mass of the rod over [0, 1] with density function p+1.

Explanation / Answer

A) consider interval a < = x < = b

variable density rho(x)

c is fulcrum

divide the part on the right into N pieces

length of part on the roght = b - c

length of each piece = (b - c)/N

coordinate of ith piece = c + i(b - c)/N

then

for ith piece to the right, moment = force * distance from l;ever

also rho is considered to be weight per unit length

moment = rho(c + i(b - c)/N)(i(b - c)/N) sum from i = 0 to i = N

as N -> inf this becomes

M = integral(rho(x)x*dx)

b) M = -integral(rho(x)(c - x)dx) x from 0 to c

c) for perfect balancing at C

Mac + Mcb = 0

integral(rho(x)x*dx) from x = c to x = b = integral(rho(x)(c - x)dx) x from 0 to c

d) rho(x) = x^2 + 1

(b^4/4 + b^2/2) - (c^4/4 + c^2/2) = c(c^3/3 + c) - c^4/4 - c^2/2

3b^4 + 6b^2 = 4c^4 + 12c^2

a = 0

b = 1

3 + 6 = 4c^4 + 12c^2

4c^4 + 12c^2 - 9 = 0

c^2 = 0.62132034355

c = 0.78823 m

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