$[$D$]$ A toy top is in the form of a cone with a circular top of radius $15[cm],$ and height $45[cm].$ The cross$-$section of the top when the point is at the origin is given by $y=3|x|.$ The moment of inertia for the top is $233[k{g}^{**}{m}^2].$

Calculate the uniform density $[k\frac{g}{m^3}]$ from the moment of inertia and the appropriate integral.

$[8$ points$]$

$\frac{3n}{h}{x}^{2}y(r(y){)}^2\frac{q_{r^2}^2}{h^2}{x}^2{\displaystyle \underset{0}{\overset{h}{}}}\frac{h^3}{3}\frac{a_n^2}{r^2}\frac{h^3}{3}$

$(\frac{3r}{h}x{)}^2\frac{4r^2}{h^2}{\displaystyle \underset{0}{\overset{h}{}}}{x}^2$

$3r^{2}h$

The top is accelerated to a spin rate $w$ by pulling a string wrapped around the $15[cm]$ radius of the cone. Rube decides to run this string over a pulley of mass $50[g]$ and radius $2[cm]$ with its axis horizontal. The end of a string is attached to a $100[g]$ mass which is released from rest.

Calculate the acceleration of the hanging mass when released. Label the diagram to define the variables you use. $[10$ points$]$

Calculate the angular velocity of the top when the mass has dropped $20[cm].[6$ points$]$