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14   PRINCIPLES OF ELEMENTARY DYNAMICS

If mass be expressed in slugs and distance in feet, the moment of inertia will be expressed in British engineering units sometimes called slug-feet2.

The moment of inertia of a body with respect to one axis has not the same value as the moment of inertia of the same body with respect to a different axis. The moment of inertia of a given body with respect to a given axis is numerically equal to the mass of a 'body which, if concentrated at unit distance from the axis of rotation, would require the same torque as the original body to produce the same angular acceleration. For any rigid body revolving about any axis fixed in space, the moment of inertia with respect to that axis is a constant quantity quite independent of both the speed of rotation and the torque acting.

The distance from the axis of rotation at which the entire mass of a body might be concentrated without altering the moment of inertia of the body with respect to that axis, is called the radius of gyration, or swing radius, of the body about the given axis. If the entire mass M of the body were at the distance k from the axis, the moment of inertia of the body with respect to this axis would be

TRANSLATION AND ROTATION   15

12. Values of the Moment of Inertia of Certain Bodies. - The moments of inertia of a body, having regular geometrical shape, can be computed, but the moments of inertia of a body of irregular shape are usually most easily determined by experiment. The experimental determination is usually made by comparison with a body of known moment of inertia. For such comparisons cylinders and rings of known dimensions are convenient.

It is shown in books on Mechanics that the moment of inertia of a uniform right solid cylinder of mass m and diameter d, about its geometric axis, is

s md2   (22

whereas about any axis parallel to the geometric axis and dicta p from it, the moment of inertia is

8 md2 + mpg   (23)

If the cylinder has a length x, the moment of inertia about an axis through the center and normal to the length is

m d2

+ X2]

16 12)   (24)

K = Mk 2   (21)

In this equation, k is the radius of gyration of the body.

Experiment. - Clamp the frame of a bicycle in an upright position with the front wheel off the ground. With the wheel not spinning, apply a torque to the handle-bars so as to impart to the front wheel an angular acceleration about the handle-bar post as an axis. Now set the front wheel spinning and apply a torque to the handle-bars as before. Note that the torque to produce a given angular acceleration is much greater when the wheel is spinning than when it is not.

The torque required to impart to a spinning body a given angular acceleration about an axis perpendicular to the spin-axis is greater than the torque required when the body is not spinning. The ratio of the torque applied to a spinning body about an axis perpendicular to the spin-axis, to the angular acceleration thereby produced about the torque axis, is sometimes called the dynamic moment of inertia of the spinning body with respect to the torque-axis. The magnitude of the dynamic moment of inertia of a body with respect to the torque-axis depends upon the angular momentum of the part of the body that is spinning, with respect to the spin-axis.

whereas about an axis coinciding with the diameter of one end, the moment of inertia of the cylinder is

mL16+ 3J   (25)

The moment of inertia of a ring or right hollow circular cylinder of outer diameter do and inner diameter di, with respect to the geometric axis, is

8 m[d 2 + die]   (26)

If the moment of inertia of a body of mass m about an axis through the center of mass of the body is KK, then the moment of inertia about any parallel axis distant p from the first axis is

Kp = K, + mpg   (27)

13. Axes of the Principal Moments of Inertia of a Body. - The moment of inertia of a sphere about an axis through the center of mass equals the moment of inertia about any other axis through the same point. The moment of inertia of a right cylinder about an axis through the center of mass and perpendicular to the length of the cylinder is greater than the moment of inertia about

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