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Click here to see a close-up view of the second part of the experiment

Click here to see a video of the second part of the experiment

On July 1, 1940, the Tacoma Narrows bridge was opened to traffic. The main span was 2800 feet long, 39 feet wide and supported by eight foot tall stiffening girders. Early on the morning of November 7, 1940, winds of about 40 mph velocity set the main span into torsional oscillation in 2 segments with a frequency of 14 vibrations per minute and a double amplitude of 70 degrees. The bridge collapsed at 11:00 a.m.

When a mass, m, is suspended from a spiral spring of force constant k, the period of oscillation is given by:

Using the analogy developed earlier in the course between linear and angular motion, we expect a similar expression to describe the period of an object in rotational oscillation as a result of some restoring torque. Just as the moment of inertia, I, replaces the mass in our new equation, so will a torsion constant, L_o, replace the force constant. The torsion constant is the torque required to twist the system through one radian of angle. Thus, the new equation for torsional oscillation is:

The key to this experiment will be the realization that the torsion constant will depend upon the dimensions and material of the chosen support rod, and also upon the calculation of moment of inertia for a variety of shapes and axes.

When you have completed this experimental activity, you should be able to: (1) write and apply the formula relating the period of a torsion pendulum to its moment of inertia and its torsion constant; (2) calculate the moment of inertia of a cylinder (either solid or hollow) about the cylinder’s axis; (3) use the parallel axis theorem for moments of inertia; (4) conduct dynamic experiments to determine the torsion constant of a rod; (5) name the physical properties of a rod which contribute to its torsional constant; and (6) relate the torsional constant of a rod to the shear modulus of the material of which the rod is formed.

Begin by measuring and recording the mass and physical dimensions of the disk, ring, and "inertial cylinders" at your work station. Be careful when measuring the mass of the disk and ring. Do not drop them on the balance. Then, using the formulas found in your textbook for moments of inertia for various shapes, calculate and record the moment of inertia for each item about its cylindrical axis. Use three significant figures throughout.

Take the brass rod. Measure carefully the diameter of the rod and the length between the collars. Place the disk securely on one end, and clamp the other end in the support on the wall. Twist the disk slightly (about 5 or 10 degrees) and release, and notice the oscillation of the disk. A small reference mark on the front of the disk may help you follow the oscillating motion better. With your stopwatch, time 50 oscillations of the disk, and record the time. Repeat the timing twice more, recording your timing data carefully and checking for consistency. Next, place the ring on the disk, and repeat the timing observations as before, carefully recording all trials. Finally, add the two cylindrical masses to the disk and ring system, and repeat the timings as before.

Record with the timing data the value of the system’s moment of inertia. Since moment of inertia is a scalar, it will simply be additive as more items are added to the system. However, when the cylinders are placed on the disk, they do not rotate about their own axis, but rather rotate about the axis of the disk. This changes (increases) the moment of inertia of these cylinders, and this change must be accounted for. It can be shown that for an object rotating about an axis parallel to an axis through the object’s center of mass, the moment of inertia is given by:

where I_o is the moment of inertia about the object’s center of mass, m is the mass of the object, and x is the distance between the object’s axis and the actual rotation axis. Carefully measure the distance from the axis of the rod to the axis of the cylinder and compute the moment of inertia of the cylinder as it rotates on the disk.

Now, plot a graph of T² versus I for the experiment. This plot should be a straight line, with its slope equal to 4p ²/ L_o. Determine the slope of the line and calculate from it the value of L_o.

Another value of L₀ can be found from the torsion apparatus, and used for comparison. Take the rod to the torsion apparatus, measure the radius of the wheel (R), and measure the angular twist of the wheel when loads of 250 g, 500 g, 750 g, and 1000 g are placed on the weight pan. The torsion constant, L₀ is given by:

where q is in degrees and the 57.3 factor is used to convert to radian measure. Use the average of the 4 values as the "best" value of L₀, and use percent difference to compare this value of L₀ to the one obtained from the graph.

Finally, we can compute the shear modulus for the brass rod from our knowledge of its torsion constant and its dimensions. Given that:

where r is the radius, l is the length, and S is the shear modulus for the rod, you should compute the value of the shear modulus. Compare this value with the value found for brass in either the Handbook or your textbook.

Report your value for L₀. How close is it to the standard value? Comment on the accuracy of this experiment. Report your three values for inertia and their corresponding periods. How does inertia affect the period. What additional conclusions can you draw from this lab?

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