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Liferider
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Homework Statement
Find the mass and the coordinates for the center of mass of a thin wire formed like a quarter circle.
Homework Equations
Circle equation: x2+y2=r2
Mass density: rho=x+y
The Attempt at a Solution
I know that: x2+y2=cos2(t)+sin2(t)=1
This leads to: x2+y2=r2(cos2(t)+sin2(t))=r2
x=r*cos(t)
y=r*sin(t)
The quarter circle is described by the vector equation:
r(t)=r*cos(t)i+r*sin(t)j; 0<t<1/2*pi
The density function becomes:
rho=x+y=r*cos(t)+r*sin(t)=r(cos(t)+sin(t))
The line integral that I try to use: (C is the quarter circle wire)
M=∫[itex]_{C}[/itex](rho(x,y)*ds)=∫[itex]^{1/2*pi}_{0}[/itex](rho(x,y)*r'(t)*dt)
=r2∫[itex]^{1/2*pi}_{0}[/itex](cos2(t)-sin2(t))*dt
This ends up in M=0... FFFUUUU! The right answer is supposed to be 2r2
I can't seem to find the right solution to this... please help.