Answer
a) $\frac{M}{a+1}$
b) $x_{cm}=\frac{ML}{2+a}$
c) These are not surprising, for they are the equations for when a rod is uniform, which is the case when a=0.
Work Step by Step
a) We know that the mass is the integral of the equation over the length of the rod:
$m = \int_0^L \frac{Mx^a}{L^{1+a}}dx=\frac{M}{a+1}$
b) $x_{cm} = \int_0^L x \frac{Mx^a}{L^{1+a}}dx$
$x_{cm} = \int_0^L \frac{Mx^{1+a}}{L^{1+a}}dx$
$x_{cm}=\frac{ML}{2+a}$
c) These are not surprising, for they are the equations for when a rod is uniform, which is the case when a=0.
