18.01 Section, November 16, 2015
Section leader: Eva Belmont (ebelmont@mit.edu, E18-401B)
.
Physical applications of integrals: center of mass and work
Review
Rb
• Mass: a ρ(x)A(x) dx (where ρ is density – which is assumed to be constant on a slice – and A(x) is
cross-sectional area)
Z b
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• Centroid (center of mass) has x-coordinate =
xρA(x) dx.
mass a
Rb
R
• Work = force × distance. Infinitesimal version: W = a F (x) dx (or W = dW split up some other
way – see problems 2 and 3).
Problems
1. Write an expression for the center of mass of a sphere of radius 1 centered at the origin, with density
ρ(x) = 1 + x.
You do not actually have to evaluate the integral(s).
2. A cylindrical tank has radius 1 m and height 2 m and is filled with water. I get a pump and pump
all the water out over the top of the tank. Water has a density of 1000 kg/m3 .
(a) What is the force exerted by gravity on the cylindrical slice of thickness dz at height z?
(b) What is the work required to pump out just this slice?
(c) How much work is done to pump out the entire tank? (i.e. to raise all the slices to the top)
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3. A metal chain of weight-density 2 lb/ft and length 5 ft is hanging from the ceiling. I grab the bottom
of the chain and raise it to the ceiling.
(a) What is the force exerted by gravity on an infinitesimally small “link” of length dx?
(b) What is the distance travelled by a link that starts at height x (say distance is measured
downward, with ceiling height = 0 and the bottom of the chain at height 5 ft).
(c) How much work have I done on a link of length dx that starts at height x?
(d) What is the total work done on the chain?
Just write the integral; don’t bother to evaluate it.
4. A spring has natural length of 12 in, and a 45-lb force stretches it to 15 in. Find the work done in
stretching it from 15 to 19 inches.
Recall that the spring force is F (x) = −kx for some constant k, where x is the distance from the equilibrium point.
5. Bonus problem: Redo problem 3, under the assumption that this is instead going on at a planetary
scale, where a link of height x (still measured from the “ceiling”) experiences a gravitational force
c · length
of (7−x)2 for some constant c (this corresponds to the assumption that the bottom of the chain
is 2 away from the center of gravity – we’re dispensing with units).
No need to evaluate the final integral.
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