Parallel axis theorem

Simplifying, the area moment of inertia for the 2-teeth cutter, which is displayed, e.g., in Fig. 16, can be calculated analytically by taking two half circles and using basic formulas as given in, e.g. (Beitz et al. 1995), using the parallel axis theorem and coordinate transformations for rotations. 

This expression is referred to the parallel axis theorem. Notice that the value md is always positive it means that the MI relative to the symmetry axis 4 has a minimum value. 

The rod MI is 4 = mJ I4 (because the rod mass is 3nii and the axis passes throngh the center of the rod). The MI of the hoop is the sum of the MI of the hoop itself (first item) and the addition from the parallel axis theorem (second item) ... 

Solution The MI of the combined body can be calculated according to the formulas presented in Section 1.3. Choose an x-axis directed along the rod with its origin at point 0. The overall MI of the whole body relative to the x-axis is the sum of the composite details 4 = 4i + 42, where part 1 is the rod and 2 is the disc. In order to find 4i and 42 we should use the theorem on parallel axis (1.3.48). The rod s MI can be given by the expression... 

As discussed in detail in [10], equivalent results are not obtained with these three unitary transformations. A principal difference between the U, V, and B results is the phase of the wave function after being h ansported around a closed loop C, centered on the z axis parallel to but not in the (x, y) plane. The pertm bative wave functions obtained from U(9, <])) or B(0, <()) are, as seen from Eq. (26a) or (26c), single-valued when transported around C that is ( 3 )(r Ro) 3< (r R )) = 1, where Ro = Rn denote the beginning and end of this loop. This is a necessary condition for Berry s geometric phase theorem [22] to hold. On the other hand, the perturbative wave functions obtained from V(0, <])) in Eq. (26b) are not single valued when transported around C. 

Fill in the missing steps in the derivation. Hint In finding Eq. (1), he has started with the equation for the length of an element on the surface of the cylinder, parallel to the axis, and then used the binomial theorem to perform the necessary integration. Similarly, to obtain Eq. (4) from Eq. (3), he has used the binomial theorem. In both cases this is the form of (l + x) = l+ x + other terms. When x is small, the other terms may be neglected, which he has done here. 

This is the Fourier slice theorem, which states that the Fourier transform of a parallel projection of an object taken at angle 0 to the a axis in physical space is equivalent to a slice of the two-dimensional transform F(u, v) of the object function f(x, y), inclined at an angle 0 to the u axis in frequency space (Fig. 26.16). 

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