Mass of the body M, Cartesian reference frame.

Question

Oxyz is a Cartesian frame of reference with unit base vectors, $i,j$ and $k$. A rigid body $V$, of uniform density $p$, is bounded by the surfaces $y=(1-x^2)^{(1/2)}, z=0, y=0$ and $z=1-y$ If the mass of the body is $M$, show that $M= \frac{p(3\pi -4)}{6}$

Answer 1

In the $x y$ plane, $$y=(1-x^2)^{(1/2)}$$

is $$x^2 + y^2 = 1 $$

... the circle of unit radius centred at the origin.

So the body is a right prism along the $z$ axis with a semi-circular section.

• To find its volume, multiply its cross-sectional area by its height.
• To find its mass, multiply this by its density.