INERTIA:

Let us take a moment to consider rotational inertia (moment of inertia). There are three useful tricks for calculating I without integrating. One of them, the parallel axis theorem, is discussed in most physics texts and will not be derived here, but we will use it. Another theorem, useful when the axis is not through the center of mass, involves the distance to the center of percussion (sweet spot). Lastly, if one knows I for rotations about two perpendicular axes (a rectangular solid, for example) one can rotate to another axis in the same plane.

The center of percussion is the point at which a collision will not cause forces on a constrained axis. Baseball players are aware of the fact that the bat stings their hands if the ball is hit too close to their hands or too close to the end of the bat; the best point is the center of percussion, a distance R_cp from the center of the hands. Consider a force applied to the center of percussion. If no other force acts on the body, the rotation will be about a point a distance R_cp away from the force (or line of force if it is not perpendicular to the body. So

R_cpF = Ia . (equation 1)

It is a remarkable fact that the total torque with respect to the center of mass is equal to I_cm a even if the rotation is not about the center of mass! If F is applied to any other point, we would need to include torque due to a force of the pivot. But in this case we apply F to the center of percussion, so there is no force at the pivot, and
xF = I_cm a , where x is the distance to the center of mass.

Also, the parallel axis theorem is I = I_cm + mh², where h is the distance from the pivot to the center of mass. The distance from F to the cm is R_cp - h, and I_cm is I - mh² so

(R_cp-h)F = (I-mh²)a . (equation 2)

If equation (2) is divided by (1), a little algebra yields the result

I = mhR_cp (equation 3)

which is nice, because h and R_cp are both easily found experimentally. h is the distance to the center of mass or the balance point, so that's easy. It can be shown that R_cp is the length of a simple pendulum which has the same period as our object's physical pendulum period. The most precise way of finding R_cp experimentally would make use of this fact. ("Physical pendulum" is physics-speak for any object that swings back and forth like a pendulum. We like to reserve the name "simple pendulum" for a ball on a string for the case of the string's mass negligible compared to the ball and the ball's radius negligible compared to the length of the string.)

Equation 3 needs to be touted a little, because although it may be the best way to find I experimentally, you probably won't find it in your text.

Now consider a two-dimensional case in which we know I about the x axis (I_x) and about y (I_y) and suppose the integral of xydm = 0, so that it can be rotated about x or y with no tendency to vibrate (no torque on the axis). Now consider rotated uv axes in the xy plane. If the angle between x and u is q then it can be shown that if point p has coordinates (x,y), then its u coordinate = xcosq + ysinq . Now I_u = v² dm. But u² + v² = x² + y², the square of the distance from p to the origin in both systems. Solving for v², we find

v² = x² + y² - x² (cosq )² -y² (sinq )² - 2xysinq cosq . (equation 4)

Simplifying with the aid of couple of trig identities, we find that

v² = x² (sinq )² + y² (cosq )² - 2xy(sinq )². (equation 5)

This is valid for all points. If we multiply by dm and integrate, and assume the integral of the last term is zero, as discussed above, we obtain

I_u = I_y (sinq )² + I_x (cosq )². (equation 6)

For an example of the use of this theorem, consider a square plate. The inertia with respect to axes through the center of opposite edges is I_x = I_y = (1/12)mL². The squares of sin and cos of 45^o are both 1/2, hence I about an axis through opposite corners is (1/12)mL² (1/2) + (1/12)mL² (1/2) = (1/12)mL² , which certainly is surprising. Who woulda thunk that they would be the same?

Now you have so much inertia that you won't be able to go back to the main physics page or to the angular acceleration page.

Other main physics pages:

fluids, heat, electricity and magnetism
vibrations and waves
quantum

F. Gram, Cuyahoga Community College, Cleveland, OH 44115.
comments, questions: fredrick.gram at tri-c.cc.oh.us , but remove spaces and replace “at” with the at symbol. (Spammers have software that can automatically get email addresses listed.)