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3D moments - what am I doing wrong?

  1. Jul 7, 2017 #1
    1. The problem statement, all variables and given/known data
    A 200N force is applied to the handle of the hoist in the direction shown. The bearing A supports the thrust (force in the direction of the shaft axis), while bearing B supports only radial load (load normal to the shaft axis). Determine the mass 'm' which can be supported Assume neither bearing to be capable of supporting a moment about a line normal to the shaft axis.

    3186413086974b0fb107185eaf343ad7.png

    2. Relevant equations
    [tex]\Sigma M = 0[/tex]
    [tex]\Sigma F = 0[/tex]
    [tex]M = r \times F[/tex]
    3. The attempt at a solution
    Before I begin, I should start by saying that I realize the solution to this problem is very easy and that the way I'm doing it is needlessly complex. However, I do not understand why my method does not work.

    I start by defining a coordinate system and an origin. This system uses the axes depicted in the figure, with the origin set at the point of connection of the handle arm and the shaft. Positive is defined as into the page, up, and right for the x, y, and z components, respectively. I will sum moments around the origin (connection to shaft). I will also use meters instead of millimeters.

    Next, I decompose the 200 N force into it's respective components.
    [tex]F_x = -70.71N[/tex]
    [tex]F_y = -173.2N[/tex]
    [tex]F_z = -70.71N[/tex]

    As a vector,
    [tex]<-70.71,-173.2,-70.71>[/tex]

    Next, a position vector, 'r' from the origin to the point of application of the force is required.
    [tex]<0.25,0,0.075>[/tex]

    The moment about the origin due to the 200 N force is given by:
    [tex]M_F = r_O \times F[/tex]

    The resultant vector is
    [tex]<13,12.37,-43.3>[/tex]

    Next, I compute the moment about the origin due to the weight of mass 'm'. The position vector from the origin to the point of application is:
    [tex]<-0.1,0,-0.35>[/tex]

    The weight vector 'w' is:
    [tex]<0,-9.81m,0>[/tex]

    Finally, the moment about the origin by the weight is given by:
    [tex]M_w = r_w \times w[/tex]

    This produces the vector:
    [tex]<-3.43m,0,0.981m>[/tex]

    The sum of the moments must equal zero. So,
    [tex]<13,12.37,-43.3> + <-3.43m,0,0.981m> = <0,0,0>[/tex]

    Right away, two problems arise. Firstly, the 'y' or 'j' components result in a non-solution. 12.37 + 0 does not equal 0. Secondly, two equations for 'm' exist, and they are not redundant. In this case, 'm' is over-constrained.

    I'm new to 3D statics and the text I'm using is strangely terse with respect to using this method. I should note that I've used this method on several problems before and have obtained the correct solution. I'm not sure what I'm doing wrong here.

    Any help would be greatly appreciated!
     
  2. jcsd
  3. Jul 7, 2017 #2

    haruspex

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    No. The forces on the bearings can have moments about the origin - just not a moment about the z axis.
     
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