alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

In the hydrogen atom the lighter electron revolves

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Unformatted text preview: rotation as in positive for counter-clockwise and negative for clockwise. y IY By applying the Pythagoras IP Theorem with P as the origin: IX IP = IX + IY . P x 194 11 Example 1 1: Consider a thin rod AB of mass M of uniform density and length L along the x-axis. The rod is free to rotate about the y-axis passing through the center of mass and perpendicular to its length. y A 4321 4321 4321 4321 − L/ x 2 dx B x +L/ 2 Since the mass M is of uniform density we have: mass per unit length = M/ L . So an element of the rod has mass dM = M/ L . dx element The moment of iner tia of this element of the rod about the y-axis is dependent moment inertia element on the distance x from the y-axis. Therefore: dIY = (M/ Ldx) . x2 = M/ Lx2dx . The moment of iner tia of the entire rod about the y-axis is got by continuously moment inertia summing up or integrating over the range of x from − L/ to +L/ . 2 2 IY = ∫ dIY +L/ = 2 ∫ M/ Lx2dx −L = /2 +L/ 2 M/ L [x /3 ] 3 − L/ 2 = M/ 3L[(+L/ )3 − (−L/ )3] 2 2 2 IY = ML/12 195 If we now shift the rod so that the axis of rotation (y-axis) passes thru one end of the rod, what is the moment of iner tia IY ? moment inertia y y 4321 4321 4321 4321 4321 0 dx B +L x A −x −L 4321 4321 4321 4321 4321 A dx B 0 2 In both cases IY = ML/3 A position vector has both magnitude (distance) and direction. The direction may be positive or negative depending on which side of the axis the rigid body is located. It is impor tant to note that the measur e off the tendency to rotate is defined measure o tendency rota otate m easur independent of the side of the axis the mass is located. The orientation is o rientation the distance of the mass from the axis or point of rotation. We observe that the greater the distance of the mass from the axis or point of rotation, the greater the moment of iner tia . Hence the easier it is to rotate moment or turn the rigid body and so lesser energy is required. If the rigid body happens to be in rotational motion about...
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This note was uploaded on 11/29/2012 for the course PHYSICS 105 taught by Professor Tamerdoğan during the Fall '09 term at Middle East Technical University.

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