Question
Asked Oct 16, 2019

A lamina occupies the region inside the circle x^2+y^2=16 but outside the circle x^2+y^2=64 The density at each point is inversely proportional to its distance from the orgin.

Where is the center of mass?

check_circleExpert Solution
Step 1

Given:

The density at each point is inversely proportional to its distance:

k
p=
p Density
k= Proportionality constant
In polar form:
y=rsin
k
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k p= p Density k= Proportionality constant In polar form: y=rsin k

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Step 2
For the circle:
x2y16y
cos20+2 sin 0= 16rsin0
(i)
r 16sine
For the circle:
cos20+2 sin2 0 = 64
(ii)
r 8
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For the circle: x2y16y cos20+2 sin 0= 16rsin0 (i) r 16sine For the circle: cos20+2 sin2 0 = 64 (ii) r 8

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Step 3
From (i) and (ii):
8srs16sine
Solving (i) and (ii)
8 16sin0
1
sine
2
(
л 5л
5л
6 6
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From (i) and (ii): 8srs16sine Solving (i) and (ii) 8 16sin0 1 sine 2 ( л 5л 5л 6 6

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