(Solved): (a) A non-uniform rod of length l is shown in Figure 4(a). The mass density varies linearly from ...

(a) A non-uniform rod of length $l$ is shown in Figure 4(a). The mass density varies linearly from ${?}_0$ at end $A$ to $1.4{?}_0$ at end B. You are given that the mass of the rod is $M=1.2{?}_{0}l$. Show that the centre of mass of the rod is $\frac{19}{36}l$ from end $A$. (b) Now end B of the non-uniform rod is attached to a uniform sphere whose radius is $0.1l$ and the whole structure is released from the vertical position as shown in Figure 4(b). The sphere has the same mass as the rod and the whole structure rotates clockwise as shown. Determine the angular velocity (in terms of $l$ ) of the structure when the structure is horizontal. (Hint: the moment of inertia of the rod about its centre of mass $(\mathrm{CM})$ is $0.0724M{l}^2$.)