Help with some Moments problems

[vdfortd]

Homework Statement

The direction cosines of the force F are cos \vartheta_x= .818, cos\vartheta_y= .182, cos\vartheta_z= -.545. The support of the beam at O will fail if the magnitude of the moment of F about O exceeds 100 kN-m. Determine the magnitude of the largest force F that can safely be applied to the beam.

In the drawing, the give the length of the beam, which would be the "r" variable, as 3 m. So r=3 m.

Also, the dots between variables stands for the dot product, and the X between variables stands for the cross product.

Homework Equations

|M_p|= D|F|
|M_p|= r X F <-- Not sure if this one applies here.
F=|F|e_f with e_f being the unit vector of F

The Attempt at a Solution

Our professor gave us that the direction cosines are the unit vector, so e_f= .818i +.182j -.545k.

They said that the magnitude of the Moment could not exceed 100 kN-m, so I set up the equation 100=D|F|. If D is the perpendicular line from O to the force, than the length is r, which as stated above = 3 m.

So now I have:

100=3|F| or 100/3 = |F|

I know that the magnitude of F = the square root of (x² + y² + z²)

When I do the math I just get that F = 33.33 when the answer is supposed to be 58 kN.

Homework Statement

Use Equations 4.5 and 4.6 to determine the moment of the 20-N force about (a) the x axis, (b) the y-axis (c) the z axis.

Gives me the position (7, 4, 0)m and the force vector 20k (N)

Homework Equations

Equation 4.5 : M_L= [e .(r X F)]e

Equation 4.6 : e.(r X F) = the determinate

The Attempt at a Solution

Using the equation M_L= [e .(r X F)]e. I can do the r X F part. I did that and got 80i -140j +0k. When when I have to dot that with e, which is a unit vector, i don't know where to get that unit vector from.

I can't do the unit vector of (7,4,0) because it doesn't come out right.

The answers are a=80i b= -140j and c=0

Thanks for any and all help.

 

[alphysicist]

Hi vdfortd,

I'm having a bit of trouble visualizing the situation for the first problem.

Homework Statement

Use Equations 4.5 and 4.6 to determine the moment of the 20-N force about (a) the x axis, (b) the y-axis (c) the z axis.

Gives me the position (7, 4, 0)m and the force vector 20k (N)

Homework Equations

Equation 4.5 : M_L= [e .(r X F)]e

Equation 4.6 : e.(r X F) = the determinate

The Attempt at a Solution

Using the equation M_L= [e .(r X F)]e. I can do the r X F part. I did that and got 80i -140j +0k. When when I have to dot that with e, which is a unit vector, i don't know where to get that unit vector from.

I believe the unit vector in that equation is the unit vector that is in the direction of the axis. So for part a, \hat e would be the unit vector in the direction of the x axis, etc.