Moments Under Rotation



This applet illustrates how the area moments of inertia and the product of inertia vary under rotation. The area shown is a circle that is rotated about the origin. The circle has a radius r1 and it rotates at a radius r2. The moments are all about the axes or origin. Values shown include the moment with respect to the x-axis,, the moment of inertia with respect to the y axis, the product of inertia, and the polar moment of inertia,, about the origin. The calculations are shown below. The right graph shows the relation between the moment of inertia and the product of inertia. The moments of inertia are shown on the abscissa ( x-axis) vs the product of inertia on the ordinate (y-axis). Points are added as you rotate by changing the angle . Note the scale changes as the radii are changed. Checking the box will plot the points (Iyy,-Ixy) instead of (Iyy,Ixy) and add a title the resulting figure. To clear the right graph change either radii on the left graph.


Rotate the area by moving the slider or selecting the play button. What shape appears in the right side graph? How does the shape change when you change the radii on the left graph? How does the axis scale vary when you change the radii on the left graph? How does the angle on the right change compared to the angle on the left? How would the rotation angle change if you rotated the axes instead of the circle? What happens when r2 is set to zero? How does the polar moment of inertia vary under a rotation? Do you notice a relation between the point and the point? Check the (Iyy, -Ixy) box and notice what happens to the relation between the point and the point? How would you calculate the radius r3 given the moments of inertia and the product of inertia at any rotation angle? How would you calculate the rotation angle, , where Ixy is zero given Ixx, Iyy and Ixy for a shape? Note: The observed behavior under rotation is the same for any shape.


Moments of Inertia Moments of inertia about the circle centroid from tables: Apply the parallel axis theorem: Some trigonometric identities to note to help explain Mohr's circle results: and Product of Inertia For a circle the product of inertia about the center is always zero due to symmetry. Apply parallel axis theorem for product of inertia: Another notable trigonometric identity: Polar Moment of Inertia From Pythagorean theorem and polar moment definition Combining the above formulas for moments of inertia and noting the trigonometric identity gives