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APL100, STANDAR MOMENT OF INERTIAS - Coggle Diagram

- - - - the slope leads to incerease in the effective coeffienct of friction
  - - - width goes to zero use of the def of a derivative
- - - - ESPECIALLY WITH FRICTION AND NORMAL REACTION
    - - KEEP IN MIND THE EXPERSSION INVIOLVES ANGUKLAR MOEMNTUM ABOUT CENTRE OF MASS AND THE CROSS PORODUCT TERM OF RCA AND VCA
      - IT IS NOT IW only
  - - - time derivative
      - integrated
    - - write the kinetic enrgy easily
      - no slip cylinder and rod
    - - w is same
    - - EXAMPLE
        
        LENGTH IS TWICE THE BREADTH
        
        OR MASS OF ONE BODY IS TWICE THE OTHER
    - - becuase forces paas through the com
      - w is constant
  - - - FOR A POINT ON A RIGID BODY OR ITS EXTENSION YOU MAY WRITE H AS IW but not for the concident point look at the exercise questio on pagr 415
- - - - FORCE PERPENDICULAR COMPONENT TO COPLANR FORCES
      - ALL THE IDEAS IN 2D TRUSSES IF YOU FIND COPLANAR FORCES
  - - - ALL THE FORCES HAVE TO BE ZERO A
  - - - BASICALLY IN THIS CASE YOU CANNOT NOT BE ABLE TO FIND ALL THE 4 FORCES
        
        BUT YOU MAY BE ABLE TO FIND SOME OF THE FORCES
      - ESPECIALLY IF THREE FORCE ARE CONCURRENT
    - - SO CAREFULLY CHOOSE THE POINTS
        
        THEY NEED NOT ONLY BE THE POINTS FROM WHICH THE FORCES COMING IN THE FBD
    - - NOT FOLLOWING THE CONVENTION
        
        FORCES AWAY FROM JOINTS
      - NOT TAKING ALL THE FORCES ON THE HALF PART AND FORGETTING THE FORCES OTHER THAN THE UNKNOWN ONES
      - BE CAREFUL WITH THE ANGLES
      - WRONGLY MARKING THE SHEAR AND THE NORMAL FORCE
        
        NORMAL ALING THE LENGTH SHEAR IS PERPENDICUALR TO LENGTH
      - THE SIGN OF THE DISTRIVUTED FORCES OR MOMENTS
      - DONT FORGET THE JUMPS AND TAKE CARE OF SIGN THEIR
      - ALL THE FUNCTION WITH RESPECT TO SAME ORIGIN
        
        EVEN THE N(X) ,Q(X), MEANS EVEN THE FUNCTION OF DISTRIBUTED FORCES OR MOMENTS
      - DONT FORGET TO SEE THE STARTING AND THE END JUMPS
      - USE CALCULUS AND
        
        DONT JUST GO BY INTUITION
        
        LIKE IF SOMETHING WAS POSITIVE AND THEN IT IS GOING TO ZERO THAT MEANS THE INITIAL POSITIVE VALUE WAS THE MAXIMUM
      - Found forces in wrong member
      - In hurry ignore members and do mistakes in joints
    - - USEFULNESS OF THE EULERS 2ND AXIOM
        
        FIND A POINT WHERE MANY FORCES ARE CONCURRENT
      - IN A SECTION FIND ONLY THE FORCES YOU WANT NOT ALL
      - MIX METHOD OF JOINTS AND SECTIONS
        
        YOU MAY NOT NEED TO FIND ALL THE FORCES AT A JOINT TO FIND A PARTICULAR FORCE AT THAT JOINT FIRST WRITE THE RELATION BETWEEN THE FORCES ON THE JOINTS
      - Sometimes you need to take section to find the reactions from dupport
- - - - AN AXIS SUCH THAT THE UNIT VECTOR ALONG ITS DIRECTION IS AN EIGEN VECTOR OG=F THE INTERIA MATRIX
      - FINDING THIS AXIS
        
        USE THE BASIC DEFINATION
        
        GEOMETRICALLLY
        
        USING THE PLAN EOF MASS SYMMETRY
        
        normal to this plane at the point it intersects the the plane
        
        IN CASE OF BODY OF REVOLUTION
        
        about any point on the axis of revolution the axis of revolution and any line normal to it is also a PA
        
        if you get two orthoganl planes of symmetry then about the point of there intersection the normal to these planes are the PA also the the line of intersection of the planes
        
        find two independent vectors perpendicular to the axis for whcih the product of inertia are zero is sufficient
      - EXISTENCE
      - HA vector if expressed with wrt to the bases vectors as the eigen vectors of the interia matrix(the vectors along the principal axes
        
        only the three terms
      - use the basis for the inertia matrix as the eigen vectors
        
        you understand what it menas to say that write the inertai matrix wrt to a basis set
        
        can you connect the inertia matrix to an opertaor
        
        the unit vectors along the axes as a means for teh matrix representation of an operator
        
        doubt is inertia matrix operator a linaer operator
        
        can ypu connect the change of bases for the matrix repe of an operator to the change of bases when inertia matrix
        
        is inertia matrix an hermitain matrix
        
        what is the consequences if it the case
        
        if yes what are the vector spaces we are considering ???
      - HA in direction of w if only if w along one of the principal axes at A
    - - DOUBT A IS ON THE RIGID BODY ??? OR THE MASSLESS EXTENSION???
      - THE COORDINATE SYSTEM USED TO FIND THE INERTIA MATRIX MUST BE THE SAME AS THE COORDINATE SYSTEM USED TO EXPRESS THE ANGULAR VELOCITY
      - NEXT TIME YOU SOLVE A DYNAMICS PROBLEM BE AWARE OF THESE POINTS
  - - - the moemntum of inertia terms have increased
    - - special cases
        
        if you have the instanatanous axis of rotation
        
        when the angualr veoclity only about one axis
        
        when you have the inertia matrix wrt to PA
    - - without work due external moments
      - work including the work due to external moments
      - REMEMBER THE WORK DONE BY INTERNAL FORCES MAY NOT BE ZERO
  - - - then chhose / identify appropriate coordinate system
        
        find the points where the EULERS 2ND AXIOM CAN BE APPLIED
        
        APPLY THE 2ND AND THE FIRST AXIOM AND SOLVE GENERAL THEN PUT SPECIFIC VALUES
    - - BE CAREFUL DONT CHANGE THE MASS
    - - IF COMPLEX OBJECT AND ARBITARY AXIS
        
        BETTER TO FIRST FIND THE INERTIA TENSOR WRT TO SOME STANDARD AXES FOR ALL THE PARTS THEN FIND THE TOTAL INERTIA TENSOR WRT TO THE STANDARD AXIS FOR THE COMPLEX OBJECT AND THEN APPLY THE TRANSFORMATION MATRIX TO THE FINAL TENSOR
      - PARALLEL AXIS THEOREM ONLY WHEN ONE AXIS COM
      - CUT OUT OBJECTS BE CAREFUL WITH THE MASS AND THE MOMENT OF INERTIA
- - - - Frame of reference
      - coordinate systems
        
        path coordinates
        
        osculating plane
        
        radius of curvature
        
        formula in terms of veolcity acceleration
        
        in terms of the equation of the plane curve
        
        in terms of the normal acceleration
        
        in term of the parametric form of the path
        
        define it in terms of the centre of curvature
        
        unit vectors
        
        HINT : USE THEM IF YOU KNOW THE PATH ESPECIALLY IF SOMETHING IS CONSTRIANT
        
        you need to understand that to solve a problem you need to choose a coordinate systems and knowing which coordinate system to use in which problem may help to solve the problem easily
      - time derivatives
        
        notation dot use