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Language of Rotation

In Chapter 3 we discussed force, mass, and motion. A large force on a small mass will produce a large change in velocity in a given time.
Now let's consider a 10-kg bicycle wheel of diameter 1 meter and a 10-kg pipe of diameter 5 cm, both at rest (Figure 7-1). Now we want to set them spinning by giving them a twist. Which is more difficult to set spinning?


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Figure 7-1

Even though they have the same mass, the bicycle wheel has a greater moment of inertia than the pipe, and applying a twist to it will not have as great an effect as applying the same twist to the pipe. For the simple shape of a ring (or pipe) turning about a central axis, the moment of inertia is

I = MR2……(1)


where M is the mass of the object and R is the radius. Do not memorize the equation, but do remember the general rule: If two objects have the same mass, then the object with greater radius will have a greater moment of inertia and thus will be more difficult to set spinning from rest.
If an object, like a bicycle wheel, is spinning, then the period T is the time it takes for one revolution. The frequency f is the number of revolutions per unit time, so

f = 1/T……(2)


which is measured in [1/s = Hertz = Hz].
A torque is a twist, which can change the frequency at which an object is spinning. A large torque on an object with a small moment of inertia will produce a large change in its frequency of rotation. Note the similarity with the second law of motion.
Notice that the moment of inertia depends also on the axis about which an object is turning. We can twist a barbell about its central axis, or about a perpendicular axis (see Figure 7-2). The moment of inertia with respect to the perpendicular axis is greater than that with respect to the central axis, because of the greater radius.

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Figure 7-2

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