## Inertia III - rectangles & spheres

Rectangles

1.  Write down the moment of inertia of a uniform thin rod when rotated about an axis through its centre of mass which is perpendicular to the line of the rod.

2.  In consequence, write down the moment of inertia of a rectangular uniform lamina (2a 2b) when rotated about an axis through its centre of mass which is parallel to the side of length 2b. Write down the inertia the other way, perpendicular to this.

3.  Apply the parallel axes rule to write down the inertia of the rectangle when the axis is at an edge.

The perpendicular axes theorem applies to laminae (not to solids) and says that if you know the inertia about two mutually perpendicular axes in the plane of a lamina, then the inertia about the third perpendicular (so perpendicular to the lamina) is the sum of the first two. If a lamina lies in the x-y plane then Ix + Iy = Iz .

Hence, show that

4  Show that the inertia of a rectangular lamina (2ax2b) about a perpendicular axis through its centre is 1/M (a²+b²)

5  Show that the inertia of a rectangular lamina (2ax2b) about a perpendicular axis through its corner is 4/M (a²+b²)

6  Write down the inertia of a uniform cube mass M and side 2a about an axis along an edge. Repeat for a cuboid of sides 2a, 2b, 2c, for axis parallel to the c sides.

Consider a right-angled triangle with hypotenuse on y-h+hx/w=0.  Mass = area = hw/2

7  By considering strips parallel to the y-axis, show that the inertia of the lamina about the y-axis is 1/6 Mw².     Write down the inertia of the lamina about the x-axis.

8  By considering strips perpendicular to the axis as rods rotated about an end, reproduce this result.

9  Apply the parallel axes theorem to find the inertia through the centroid, G, at (w/3, h/3) and apply the perpendicular axes theorem to prove that
the inertia for such a lamina is
1/18 M (h² +w²).

10  Working from the Q9, find the inertia for axes parallel to the two drawn axes through the midpoint of the hypotenuse, M. Consequently write the inertia about the axis perpendicular to the lamina through M.

11  Take two such triangles to make a rectangle 2w x 2h and write down its inertia around M. Show that this is consistent with your answer to Q4.

Spheres

12  Show by integration that the moment of inertia of a solid uniform sphere of radius r about a diameter is 2/5 Mr².

13  Write down the moment for the same sphere as Q12 about a tangent.

14  Consider the difference between a sphere of radius r and another of r-∂r. The mass of the sphere is M = 4/3 πμ r³where μ is an appropriate measure of mass. By considering the diff-erence between these two, show that the inertia of a hollow sphere about a diameter is 2/3 Mr².

15  Write down the moment for the same sphere as Q14 about a tangent.

11 as Q7 throughout; only the meaning of M changes.