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Crummett and Western, Physics: Models and Applications,
Sec.11-2,3,4 and 14-2
Halliday, Resnick, and Walker, Fundamentals of Physics (5th
ed.), Sec. 11-6,7,8, Sec.16-6
Tipler, Physics for Scientists and Engineers (3rd ed.), Sec.
8-2
The mass of
an object is a measure of its inertia -- that is, its resistance
to a change in its state of motion. In the rotational motion of a rigid
body, the quantity that plays the analogous role depends not only on the
body's mass, but also on how the mass is distributed in space relative
to the axis of rotation. This quantity is the rotational inertia, or moment
of inertia, I. The rotational inertia in turn is often expressed
as
(1)
which defines the radius of gyration, k, about a particular axis
of the rigid body. k is a characteristic length which depends on
both the size and shape of the body, and M is its mass. The radius
of gyration of a uniform disc of radius R about its axis, for example,
is 
; this is just
another way of saying that the rotational inertia of a disc is given by
½MR2.
A physical pendulum consists of any rigid body pivoted about a horizontal axis and free to swing back and forth in the vertical plane, as sketched in Fig. 1. The distance from the pivot to the body's center of mass is s, and the mass of the body is M. The period of the physical pendulum's oscillation is given by
(2)
You'll find a derivation of this in your textbook. In (2), I is the moment of inertia about the pivot point. But using the parallel-axis theorem, we can write this as
where
(3)
is the moment of inertia about a parallel axis through the center of mass of the body and kC is the radius of gyration for the axis which passes through the center of mass. Finally, substituting (3) into (2) gives
(4)
In the present experiment, you will infer kC and s from measurements of the period of the physical pendulum swinging about different pivot points; from these, you can infer both its rotational inertia about the center of mass and the location of the center of mass.
It is obvious from (4) that the period T becomes very long for large s, and also as s goes to 0; thus there must be an intermediate value of s for which T is a minimum. It is easy to show that this occurs when s = kC, so that
(5)
Thus, by finding the minimum period of oscillation, you can determine kC directly.
Your physical pendulum is a large flat piece of wood or plastic which can be pivoted at various points along its midline, as suggested by Fig. 1.
(1) Sketch the object to scale in your laboratory notebook, and record all its dimensions.
(2) Pivot the pendulum near its end and set it swinging. The amplitude of the swing shouldn't be very large, no more than 15-20 degrees. Measure the time it takes for the pendulum to undergo 10 complete cycles of its oscillation. Trade jobs with your partner, and repeat the measurement. Now each partner take another time measurement, so that you have four independent trials of the time required for the pendulum to swing through 10 cycles.
(3) Measure the distance from the end of the object to the pivot point (distance X in Figure 1).
(4) Repeat steps (2) and (3) for several other pivot positions along the midline of the body (something like 7-12 different positions should be enough). Your primary purpose is to determine the minimum oscillation period; try to take data that will let you do so as precisely as possible.
(1) Derive Equation (5) from Equation (4).
(2) For each pivot position, calculate the mean value and standard error of the pendulum's period. Draw a graph of T vs. X, and from it determine the minimum value of the period, Tmin. (You may want to use an expanded scale, or plot transformed values, or something, to make the minimum more sharp and clear.) Use Equation (5) to calculate the corresponding value of the radius of gyration kC. From the errors in your T-values and in your graph, estimate the uncertainty of Tmin and the corresponding uncertainty in kC. (See Chapter 2 if you need to review error-propagation calculations.)
(3) Compare your final value of kC with the scale drawing of your physical pendulum. Remember that kC is the effective distance (for rotational motion) of the body's mass from its center of mass. Does the value you got appear reasonable? Comment.
(4) Choose several values of X and T for which T is substantially different from Tmin. For each, use Equation (4) and your experimental value of ks to calculate s. Make a table of X, T, s, and XC = X + s. (See Fig. 1.) XC is the position of the center of mass of the object. Calculate the mean, standard deviation, and standard error of the values of XC that you get. Indicate the position of the center of mass on your scale drawing of your pendulum.
(5) You can also locate the center of mass by determining the pivot position at which T = Tmin. From the discussion around Equation (5) we can see that, if the period has its minimum value when X = Xmin, then XC = Xmin + kC. Determine XC, and estimate its uncertainty, by this method. Which means of determining Xmin do you think is preferable? Why?
Angular Acceleration and Moment of Inertia
last update 6/97