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.

The spur gear is subjected to the two forces

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.

The spur gear is subjected to the two forces

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

 

Final Answers:

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

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

 

This question can be found in Engineering Mechanics: Statics, 13th edition, chapter 2, question 2-73.

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