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If the angular velocity is then kinetic energy of

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Unformatted text preview: ht radius = r f(x) x =h The cylinder is known as a solid of rotation . The technique used in example 8 solid to find the volume of a cylinder cannot be extended to other solids of rotation. solids Because we cannot get an accurate measure of the element of volume dV. element volume We need an accurate measure of dV. In the case of a cylinder we may take an element of volume dV to be a solid disc element of radius f(x) and thickness dx. Then : dV = π {f(x)}2.dx = πr2.dx h h h 0 0 0 volume of a cylinder = ∫dV = ∫ π {f(x)}2.dx = ∫ πr2.dx = [ πr2x] = πr2h This technique may now be applied to other solids of rotation. solids h h volume of cone = ∫dV = ∫ π {f(x)}2.dx = ∫ π(rx/ h)2.dx 0 23 = [ π(r x / 3h2)] h 0 192 0 = 1 3 πr2h / The physical entities like position , distance , speed , acceleration , area and position distance speed acceleration area volume are easy to visualize. Let us now look at an application of integration where the physical entity being measured is not so obvious. M oment of I ner tia In linear motion t he distribution of the mass (density) of the rigid body l inear does not matter. We may take the center of mass of the rigid body and treat c enter it as concentrated mass or point mass . So we have only one position vector for point orientation the rigid body. Hence the orientation of the rigid body does not affect the orienta equation of energy of linear motion. In rotational motion the orientation of rotational orientation the body does affect the equation of energy of rotational motion. orientation = distance of the individual elements of mass of the rigid body elements from a particular point or axis. elements of mass = similar segments or cross-sections of the mass, more than just a particle. Contiguous elements of mass when summed up continuously or integrated form elements the entire mass of the rigid body . To keep our calculations simple at the high school level, we assume the mass of the rigid body to be of unifor m density . unifor u niform Hence they are of equal m...
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