The spur gear is subjected to the two forces 1

The spur gear is subjected to the two forces caused by contact with other gears. Express each force as a Cartesian vector.

Image from: Hibbeler, R. C., S. C. Fan, Kai Beng. Yap, and Peter Schiavone. Statics: Mechanics for Engineers. Singapore: Pearson, 2013.

Solution:

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We will first write the components of force $F_1$. Notice how it lies on the z-y plane, meaning it has no x-component.

$F_{1y}\,=\,50\left(\dfrac{7}{25}\right)\,=\,14$ lb

$F_{1z}\,=\,50\left(\dfrac{24}{25}\right)\,=\,-48$ lb

(Note how our z-component is negative. This is because the z-component of force $F_1$ lies along the negative z-axis.)

Thus, we can write force $F_1$ in Cartesian vector form.

$F_1\,=\,\left\{0i+14j-48k\right\}$ lb

Now, we will look at force $F_2$.

We can express each component of force $F_2$ by multiplying the magnitude by each coordinate direction angle shown in the diagram.

$F_{2x}=180\cos60^0=90$ lb

$F_{2y}=180\cos135^0=-127$ lb

$F_{2z}=180\cos60^0=90$ lb

Let us now express force $F_2$ in Cartesian vector form.

$F_2\,=\,\left\{90i-127j+90k\right\}$ lb

$F_1\,=\,\left\{0i+14j-48k\right\}$ lb
$F_2\,=\,\left\{90i-127j+90k\right\}$ lb