MOMENT OF INERTIA x z y

MOMENT OF INERTIAThe moment of inertia of a rigid body is a measure of how much the massof body is spread out. It is a measure of the rigid body's ability to resistchange in rotational motion.Calculation: The moment of inertia of a rigid body, B, about an axis is⌠⎮⎮⌡r 2dmwhere: dm is an element of mass of the body,r is the perpendicular distance from the axis toany arbitrary mass element of the body (think of thePythagorean Theorem to see r^2 in the integrand),and the integral is over the mass of the whole body.See Figure 1zry∆m izxyxFigure 1: Position of an element of mass of a rigid body.Example: Find the moment of inertia of the box in Figure 2 with respect tothe x axis.zxy

Figure 2: Box (like the beam in our catapault)Dimensions of box: a in the x direction (i.e. -a/2 < x < a/2)b in the y direction (i.e. -b/2 < y < b/2)c in the z direction (i.e. -c/2 < z < c/2)Origin of coordinate system placed at center of box, which will be thecenter of gravity of the box, assuming constant density.⌠⎮I x=⎮⌡r 2dm =⌠⎮⎮⌡ρr 2dVdensity*volume = mass so dm = ρ dV=⌠⎮⎮⌡⌠⎮⎮⌡Box⌠⎮⎮⌡ρr 2dxdydzset up triple integral inCartesian coordinates to dovolume calculation=c⌠ 2⎮⎮⎮⌡− c2b⌠ 2⎮⎮⎮⌡− b2a⌠ 2⎮⎮⎮⌡− a2ρ ( y 2 + z 2 )dxdydzLimits of integration defined by where the box "lives" in the 3Dcoordinate system. Recall that we are finding the moment of inertia withrespect to the x-axis, and we need to define r, the distance between anypoint in the box and the x-axis. Referring to Figure 1, above, we knowthat r^2=y^2+z^2, by the Pythagorean Theorem.=c⌠ 2⎮⎮⎮⎮⎮⎮⎮⌡− c2b⌠ 2⎮⎮⎮⎮⎮⎮⎮⌡− b2ρ ( y 2 + z 2 a) x2−a2dydzIntegratingfirst withrespect to x

=c⌠ 2⎮⎮⎮⌡− c2b⌠ 2⎮⎮⎮⌡− b2ρa( y 2 + z 2 )dydz=c⌠ 2⎮⎮⎮⎮⎮⎮⎮⌡− c2⎛⎜⎝ρa 1 3 y3 + yz 2⎞ ⎟⎠b2−b2dzIntegratingnow withrespect to y=c⌠ 2⎮⎮⎮⌡− c2⎛ρab⎜b2⎝12+ z 2⎞⎟⎠dz⎛⎜⎝= ρabb 2= ρabc12z 1+12 3 z3( b 2 + c 2 )⎞⎟⎠c2−c2Integratingfinally withrespect to z= m (12 b2 + c 2 ) since mass = density*volume