The meshed gears are subjected to the couple moments shown. Determine the magnitude of the resultant couple moment and specify its coordinate direction angles.

#### Solution:

Show me the final answer↓

We will first express each moment in Cartesian vector form. (Don’t remember how?)

To express M_2 in Cartesian vector form, we first need to find M’, which is the component which lies on the x-y plane.

(Using M’ we can figure out M_{2x} and M_{2y})

M_x=18.79\sin30^0=-9.4

M_y=18.79\cos30^0=-16.27

**(both the x and y components are negative because both components lie in the negative x and y axes)**

We can now express M_2 in Cartesian vector form:

The resultant couple moment can be found by adding the two moments together.

M_c=\left\{0i+0j+50k\right\}+\left\{-9.4i-16.27j+6.84k\right\}

M_c=\left\{-9.4i-16.27j+56.84k\right\}\,\text{N}\cdot\text{m}

The magnitude of this couple moment is:

magnitude of M_{c}=59.86\,\text{N}\cdot\text{m}

The coordinate direction angles can be found by:

\beta=\cos^{-1}\left(\dfrac{-16.27}{59.86}\right)=105.77^0

\gamma=\cos^{-1}\left(\dfrac{56.84}{59.86}\right)=18.28^0

#### Final Answers:

magnitude of M_{c}=59.86\,\text{N}\cdot\text{m}

\alpha=99.03^0

\beta=105.77^0

\gamma=18.28^0