Question

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1. for the machine element shown, locate the x coordinate of the center of gravity. dimensions in mm

Explanation:

Step1: Divide the machine element into simple parts

Let's consider the machine element as composed of three parts: a large rectangular part, a smaller rectangular part (the vertical - like part), and a hole (which we will subtract).

1. Large rectangular part (Part 1):

• Length \(L_1=(90 - 24)\text{mm}=66\text{mm}\), width \(w_1 = 10\text{mm}\), height \(h_1=40\text{mm}\).
• Volume \(V_1 = L_1\times w_1\times h_1=66\times10\times40 = 26400\text{mm}^3\)
• The \(x\) - coordinate of its centroid \(x_1=\frac{90 - 24}{2}+24=57\text{mm}\)

1. Smaller rectangular part (Part 2):

• Length \(L_2 = 24\text{mm}\), width \(w_2=10\text{mm}\), height \(h_2 = 40\text{mm}\).
• Volume \(V_2 = L_2\times w_2\times h_2=24\times10\times40 = 9600\text{mm}^3\)
• The \(x\) - coordinate of its centroid \(x_2=\frac{24}{2}=12\text{mm}\)

1. Hole (Part 3, to be subtracted):

• Radius \(r = 12\text{mm}\), length \(L_3=10\text{mm}\)
• Volume \(V_3=\pi r^{2}L_3=\pi\times12^{2}\times10=1440\pi\text{mm}^3\)
• The \(x\) - coordinate of its centroid \(x_3 = 90 - 20=70\text{mm}\)

Step2: Use the formula for the centroid of a composite body

The formula for the \(x\) - coordinate of the centroid of a composite body \(x_{cg}=\frac{\sum_{i = 1}^{n}x_iV_i}{\sum_{i = 1}^{n}V_i}\)

$$
LATEXBLOCK0
$$

$$
LATEXBLOCK1
$$

$$
LATEXBLOCK2
$$

Answer:

The \(x\) - coordinate of the center of gravity is approximately \(41.4\text{mm}\)