Statics - 3 Dimensional Equilibrium

[sgvaibhav]

Moment about an axis - Statics - 3 Dimensional Equilibrium

Homework Statement

The member is supported by a pin at A and a cable BC. If the load at D is 300, determine the x, y, z components of reaction at these supports.
i.e Find the reactions at A and the tension in the cable BC

Homework Equations

Since it is in equilbrium.
$$\Sigma F = 0$$
$$\Sigma M = 0$$

The Attempt at a Solution

The most confusing part for me, is moment about an axis. (If possible, please tell me, how to find moment about an axis - the direct method and through the vector matrix).

Correct me, if i am wrong.

rBC = 0.3i - 0.6j + 0.2k
|rBC| = 0.7
uBC = 1/0.7 (0.3 i - 0.6j + 0.2k)

TBC = |TBC|/0.7 (0.3i - 0.6j + 0.2k)

Now
$$\Sigma Fz = 0$$
$$= -300 + Az + TBC(0.2 / 0.7)$$
Az is unknown, so to determine Az - we find moment about the x-axis of B.
This will give, Az = 0
$$\Sigma Fz = 0$$
$$= -300 + TBC(0.2/0.7) = 0$$
TBC = 1050 N.

$$\Sigma Fx = 0$$
Ax + 1050(0.3/0.7) = 0
Ax = -450 N

$$\Sigma Fy = 0$$
Ay - 1050(0.6/0.7) = 0
Ay = 900N

Thats what i could do max (i think I am right), now only moment about Y-axis w.r.t a and moment about Z-axis is left, w.r.t a again. (May and Maz).
Please tell me how to get this part, with 2-3 methods to get moment about an axis (Direct and through matrix).

Thanks a lot =)

 

[anathema]

Thank you for your post regarding moment about an axis in statics. Moment is a measure of the tendency of a force to rotate an object around a specific axis. In your problem, you are trying to find the moment about the x-axis at point B, which will help you determine the reactions at point A and the tension in cable BC.

First, let's define the moment about an axis using the right-hand rule. Imagine wrapping your right hand around the axis of rotation with your fingers pointing in the direction of the force. Your thumb will then point in the direction of the moment.

To find the moment about an axis using the direct method, you can use the formula M = r x F, where M is the moment, r is the position vector from the axis of rotation to the point of application of the force, and F is the force vector. In this case, you can find the moment about the x-axis at point B by taking the cross product of the position vector rBC and the force vector TBC.

To find the moment about an axis using the vector matrix method, you can use the formula M = [r] x [F], where [r] is the position matrix and [F] is the force matrix. The position matrix [r] is a 3x3 matrix with the x, y, and z components of the position vector rBC as its columns. The force matrix [F] is a 3x3 matrix with the x, y, and z components of the force vector TBC as its columns. Taking the cross product of these two matrices will give you the moment about the x-axis at point B.

To find the moments about the y-axis and z-axis at point A, you can use the same methods, but with different position and force vectors/matrices. For example, to find the moment about the y-axis at point A, you can use the position vector rAD and the force vector FAD, where rAD is the position vector from point A to the point of application of the force at D, and FAD is the force vector at point D. You can then use the direct method or the vector matrix method to find the moment about the y-axis at point A.

I hope this helps you understand how to find the moment about an axis in statics. Please let me know if you have any further questions. Best of luck with your homework![Your