Moment of Inertia. Moment of Inertia
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1 Moment of nertia What is a commi+ee? A group of the unwilling, picked from the unfit, to do the unnecessar. Moment of nertia When we calculated the centroid of a shape, we took the moment generated b the shape and divided it b the total area of the shape. This gave us a distance, which was the distance to the centroid of the shape Moment of nertia b ntegraion Monda, November 9, 0
2 Moment of nertia The moment of inertia is actuall the second moment of an area or mass about an ais Notice that it is not a distance, it is a moment of a moment That ma sound strange l t should 3 Moment of nertia b ntegraion Monda, November 9, 0 Moment of nertia There is reall nothing that can easil be used to describe the moment of inertia For an area, it will have units of length which is ver difficult to map to a phsical quantit Moment of nertia b ntegraion Monda, November 9, 0
3 Moment of nertia The smbol for the moment of inertia is with a subscript describing about which ais the moment is being calculated The moment of inertia about the -ais would be, about the -ais, There is also a moment of inertia about the origin, known as the polar moment of inertia designated as J O 5 Moment of nertia b ntegraion Monda, November 9, 0 Moment of nertia The moment of inertia is a phsical propert and determines the behavior of a material under certain loading and dnamic conditions Remember, we are taking the moment of the moment (the second moment) of an area about an ais Keep this in mind and ou won t have an trouble here 6 Moment of nertia b ntegraion Monda, November 9, 0 3
4 Moment of nertia The first moment of a shape about an ais was calculated b taking the moment arm to the local centroid of the shape and multipling that b the area of the shape 7 Moment of nertia b ntegraion Monda, November 9, 0 Moment of nertia The second moment will be generated in a similar manner We will take a moment arm from the ais to the centroid of the shape, square that moment arm, and multipl that product b the area 8 Moment of nertia b ntegraion Monda, November 9, 0
5 Moment of nertia For a moment of inertia about (around) a -ais, the moment arm will be measured perpendicular to the -ais, so it will be an -distance So for we would have = A 9 Moment of nertia b ntegraion Monda, November 9, 0 An Eample Consider the following figure 0 Moment of nertia b ntegraion Monda, November 9, 0 5
6 An Eample We will start with the, or the moment of inertia about the -ais Moment of nertia b ntegraion Monda, November 9, 0 An Eample To take a moment about the -ais, we will need to have a moment arm that has an - distance Moment of nertia b ntegraion Monda, November 9, 0 6
7 An Eample Again, we will begin b generating a differential area, da = d top - bottom = 3 Moment of nertia b ntegraion Monda, November 9, 0 Point to Note You must be careful that the side of the rectangle describing the differential area that does not have the differential component is parallel to the ais about which ou are taking the moment of inertia Moment of nertia b ntegraion Monda, November 9, 0 7
8 Point to Note f ou do not set up the problem this wa, the calculations are a bit different as ou have seen from the eample we did in class. 5 Moment of nertia b ntegraion Monda, November 9, 0 An Eample n this case, the height is parallel to the - ais = d top - bottom = 6 Moment of nertia b ntegraion Monda, November 9, 0 8
9 An Eample f this isn t so, the method breaks down = d top - bottom = 7 Moment of nertia b ntegraion Monda, November 9, 0 An Eample Once we have the differential area, we locate the moment arm from the ais = d top - bottom = 8 Moment of nertia b ntegraion Monda, November 9, 0 9
10 An Eample Now the second moment of this differential area will be the moment arm squared times the differential area da d = top - bottom = 9 Moment of nertia b ntegraion Monda, November 9, 0 An Eample n this eample the differential area da is the height of the rectangle times the width of the rectangle ( ) da = d TOP BOTTOM da = d d = top - bottom = 0 Moment of nertia b ntegraion Monda, November 9, 0 0
11 An Eample The moment of inertia of the differential area is the square of the moment arm times the differential area A da = d A = d = top - bottom = Moment of nertia b ntegraion Monda, November 9, 0 An Eample The moment of inertia for the complete shape,, is the sum of all the moments of inertia of the differential areas = da A ( ) = d TOP BOTTOM A = d d = top - bottom = Moment of nertia b ntegraion Monda, November 9, 0
12 An Eample Notice that we are calculating but the distances are in the -direction, be careful to remember this = da A ( ) = d TOP BOTTOM A = d d = top - bottom = 3 Moment of nertia b ntegraion Monda, November 9, 0 An Eample Evaluating the integral, we have 5 = d = = =.9 Moment of nertia b ntegraion d = top - bottom = Monda, November 9, 0
13 An Eample Using the same method, we can calculate the 5 Moment of nertia b ntegraion Monda, November 9, 0 An Eample Start b drawing the differential area = right - left d = 6 Moment of nertia b ntegraion Monda, November 9, 0 3
14 An Eample Draw the moment arm from the -ais = right - left d = 7 Moment of nertia b ntegraion Monda, November 9, 0 An Eample The second moment for this differential area is da d = right - left = 8 Moment of nertia b ntegraion Monda, November 9, 0
15 An Eample The second moment for this differential area is da ( ) RGHT LEFT d d d = right - left = 9 Moment of nertia b ntegraion Monda, November 9, 0 An Eample The for the composite area is the sum of the s for the individual differential areas 5 = d 7 5 = 7 5 = =.9 d = right - left = 30 Moment of nertia b ntegraion Monda, November 9, 0 5
16 An Eample The polar moment of inertia, J O, is the sum of the moments of inertia about the and ais J = + J = J O O O = 3.88m = = 3 Moment of nertia b ntegraion Monda, November 9, 0 An Aside Just for our information, ou are not required to know this method, ou can use a double integral to find the moment of inertia 3 Moment of nertia b ntegraion Monda, November 9, 0 6
17 An Aside The difference is how ou describe the differential area, in this case the differential area would be da = ( d)( d) = d d = 33 Moment of nertia b ntegraion Monda, November 9, 0 An Aside The second moment of this differential area about the -ais would be ( )( ) da= d d = d d = 3 Moment of nertia b ntegraion Monda, November 9, 0 7
18 An Aside As we sum the differential areas through the composite, we are integrating in two directions, and A ( )( ) da= d d A = d d = 35 Moment of nertia b ntegraion Monda, November 9, 0 An Aside 36 Since we have an, we can choose to the -direction as the inner integral and move from bottom to top TOP BOTTOM ( ) d ( ) ( ) d d ( ) ( ) d d Moment of nertia b ntegraion = d d = Monda, November 9, 0 8
19 An Aside Making the inner integration, we have d = d d = 37 Moment of nertia b ntegraion Monda, November 9, 0 An Aside Which is the same form as we had before for = d = d d = 38 Moment of nertia b ntegraion Monda, November 9, 0 9
20 Homework Problem 0- Problem 0- Problem Moment of nertia b ntegraion Monda, November 9, 0 0
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