# Calculate Resultant Couple from 3 Couples in 3D

• wannawin
In summary, the homework statement states that the magnitude of the resultant couple is 4.5012 and the direction is 0 with the +x axis.
wannawin

## Homework Statement

Replace the three couples with a single resultant couple. Specify its magnitude and direction of its axis using angles to the positive x, y, and z axes.

## Homework Equations

M = r x F (r cross F)

## The Attempt at a Solution

I'm pretty sure I want to resolve the moments into their force components, but to do that I would need to set an r. But since I have no measurements other than the angle of inclines I don't see how I can do that.

wannawin said:

## Homework Statement

Replace the three couples with a single resultant couple. Specify its magnitude and direction of its axis using angles to the positive x, y, and z axes.

## Homework Equations

M = r x F (r cross F)

## The Attempt at a Solution

I'm pretty sure I want to resolve the moments into their force components, but to do that I would need to set an r. But since I have no measurements other than the angle of inclines I don't see how I can do that.
The nice thing about couples is that they are independent of 'r'...the moment of a couple about any point is the couple itself. The couple can be represented by a vector pointing perpendicular to the plane of the couple, following the right hand rule method for its direction (+ or - ) perpendicular to the plane. Then solve the resultant couple as a vector located anywhere on the plane, but with a certain magnitude and direction.

Then I guess this question isn't that complicated at all. If you number the couples 1-3 going from left to right:
M1x=0
M1y=1.5cos20
M1z=1.5sin20

M2x=0
M2y=1.5
M2z=0

M3x=0
M3y=1.75cos25
M3y=-1.75sin25

Therefore MRx=0
MRy=4.496
MRz=-0.2265

MR has a magnitude of 4.5012

And the angles are 0 with the +x axis, 2.91 with the +y axis, and 87.1 with the -z axis (using simple trig)

Barring any issues with significant digits, I think that makes some sense since the resultant is almost vertical.

You may have your plus/- signs mixed...the y component points down, are you calling that the positive y axis? Otherwise, your work is very good.

PhanthomJay said:
You may have your plus/- signs mixed...the y component points down, are you calling that the positive y axis? Otherwise, your work is very good.

I did. Good catch and thanks a lot

## What is a resultant couple in 3D?

A resultant couple in 3D is a vector quantity that represents the combined effect of three separate couples acting on an object in three-dimensional space. It is the sum of the individual moments of the three couples and describes the overall rotational force on the object.

## How do you calculate the resultant couple from three couples in 3D?

To calculate the resultant couple, you must first find the individual moments of the three couples. Then, use vector addition to find the sum of these moments, taking into account the direction and magnitude of each moment. The resulting vector will be the resultant couple.

## Can the resultant couple be negative?

Yes, the resultant couple can be negative. This occurs when the individual moments of the three couples are acting in opposite directions, resulting in a net moment in the opposite direction. In this case, the resultant couple will have a negative magnitude and direction.

## What units is the resultant couple measured in?

The resultant couple is typically measured in newton-meters (N·m) or pound-feet (lb-ft) in the SI and US customary systems, respectively. These units represent the product of force and distance, which are the components of a moment.

## Why is calculating resultant couple important in engineering and physics?

Calculating resultant couple is important in engineering and physics because it allows us to understand and predict the rotational motion of objects. By knowing the resultant couple, we can determine the stability and equilibrium of structures, analyze the effects of forces on objects, and design efficient and safe systems.

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