moments% - ce.memphis.edu ten.pdf · 3 introduction to moments monday, february 8, 2010 tools...

"The instructor would make a good parking lot a6endant. Tries to tell you where to go, but you can never understand

him."

Moments

Monday, February 8, 2010 Introduction to Moments 2

Objec?ves

 Understand what a moment represents in mechanics

 Understand the scalar formula?on of a moment

 Understand the vector formula?on of a moment


Tools

  Basic Trigonometry

  Pythagorean Theorem

  Algebra

  Visualiza?on

  Posi?on Vectors

 Unit Vectors


Defini?on

  A moment is the tendency of a force to cause rota?on about a point or an axis


Defini?on

 When we discussed forces earlier, we looked at their tendency to cause transla'on (movement along an axis)

 Now we are looking at their tendency to cause rota'on (movement around an axis)


Defini?on

 Moment is oNen used in the same sense as torque which is also the tendency to rotate.

 We will use moment exclusively in this class


Defini?on

 The magnitude of a moment is dependent on both the magnitude of the force causing the moment and how far away the line of ac7on of the force is from the point or axis the rota7on is occurring about


A Scalar Formula?on

 One way to calculate the magnitude of a moment is use the product of the perpendicular distance to the line of ac?on of the force from the point or axis around which the rota?on is taking place and the magnitude of the force


A Scalar Formula?on

 No?ce  Magnitude of the moment  Perpendicular distance from the point or axis about which rota?on is taking place to the line of ac?on of the force

 Magnitude of the force


A Scalar Formula?on

  A two dimensional example

 We would have to take the moment of F about a


A Scalar Formula?on

  First we develop the line of ac?on of F


A Scalar Formula?on

  Then we can draw a line from a to the line of ac?on of F

  This line makes a perpendicular with the line of ac?on of F


A Scalar Formula?on

  And use the length of the perpendicular line and the magnitude of the force to calculate the magnitude of the moment


A Scalar Formula?on

  That units of magnitude for a moment are   Ft-‐lbs  N-‐m

  The order of terms doesn’t ma6er


A Scalar Formula?on

  The point about which rota?on would occur is known as the moment center

  In this example, a is the moment center


A Scalar Formula?on

  If we cannot construct a line (moment arm) which is perpendicular to the line of ac?on of the force, we can use any other line and som trig to calculate the magnitude of the moment created


A Scalar Formula?on

 We can construct a line with a length d from the moment center to the line of ac?on of the force


A Scalar Formula?on

  This line will make an angle θ with the line of ac?on of the force


A Scalar Formula?on

  Looking at the triangle formed we can state


A Scalar Formula?on

  So another way to calculate the magnitude is


A Scalar Formula?on

  The direc?on of the moment can be described in a two-‐dimensional problem as either clockwise CW, or counter-‐clockwise CCW

  By conven?on, we label CW moments as nega?ve and CCW moments as posi?ve

  You will see why when we do three-‐dimensional problems


A Scalar Formula?on

 One way to see the sense of rota?on is to think of a clock face on an old clock (definitely not a digital clock)

  The large arm is the minute hand, the smaller one is the hour hand


A Scalar Formula?on

  If something pushes the minute hand where ?me passes correctly, then it is moving the hand clockwise CW

  If something pushes the minute hand where ?me passes backwards, then it is moving the hand counter-‐clockwise CCW


A Scalar Formula?on


A Scalar Formula?on

 Now we can use the clock to determine the sense of rota?on of the moment

 We start by placing the center of the clock on the moment center


A Scalar Formula?on

 Draw the clock face so that the dperpendicular is the minute hand of the clock


A Scalar Formula?on

 Determine that if F were pulling or pushing on the minute hand would ?me be passing normally or backwards


A Scalar Formula?on

  In this case F would be causing ?me to pass backwards so the moment is CCW and therefore a posi?ve moment


Vector Formula?on

  Clockwise and counter-‐clockwise really don’t have any meaning in three dimensional problems

  Vectors make life much easier in three dimensions


Vector Formula?on

 Once again, we construct a moment arm from the center of rota?on to the line of ac?on of the force causing the rota?on

  The moment arm is nothing more than a posi?on vector from the moment center to the line of ac?on of the force


Vector Formula?on

  F is the force vector and r is the moment arm vector


Vector Formula?on

  The moment generated about point a by the force F is given by the expression


Vector Formula?on

  The cross product is the second type of vector mul?plica?on

 Unlike the dot product which produced a scalar, the cross product produces a vector

 Unlike the dot product, the order in which we write the terms is important


Vector Formula?on

 One of the most commonly made mistakes when dealing with moments in three dimensions is to put the order of the cross product in the incorrect order


Vector Formula?on

  For the dot product, the product of two like unit vectors was 1 and any other product equals 0


Vector Formula?on

  For the cross product, things are a bit more complicated


Vector Formula?on

  The cross product follows the right hand rule


Vector Formula?on

  If we have a posi?on vector r and a force vector F defined as


Vector Formula?on

 We can calculate the moment of the force about the point by taking the cross product


Vector Formula?on

  Expanding


Vector Formula?on

 Using our cross product rules for unit vectors


Vector Formula?on

 One thing to no?ce here

If we are in two dimensions (x and y) there will be no i and j components to the resulting moment. The moment will be either into the page or out of the page. Since we follow the right hand rule for all our axes, into the page would be negative and out of the page would be positive. This corresponds to CW and CCW.


Vector Formula?on

 We can also set up the cross product as a matrix


Vector Formula?on

  There are a number of ways to expand this matrix to find the solu?on, use whatever way you are comfortable with


Vector Formula?on

  Since I could never keep the signs straight, I always use what appeared to me to be a very simple technique


Vector Formula?on

  Copy the first two columns to the end of the matrix


Vector Formula?on

  Start with the i and move down and right, then leN, then up and right

 Up and right has a nega?ve sign

- *


Vector Formula?on

  So for the i coefficient we would have

- *


Vector Formula?on

 We follow the same process star?ng from j


Vector Formula?on

  So for the j coefficient we have


Vector Formula?on

  And for the k coefficient we have


Vector Formula?on

  Summing the three products

moments% - ce.memphis.edu ten.pdf · 3 introduction to moments monday, february 8, 2010 tools...

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