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FIGURE 5.1 below shows a shaped lamina - NSC Civil Technology Construction - Question 5 - 2017 - Paper 1

Question 5

FIGURE 5.1 below shows a shaped lamina. All dimensions are in millimetres. Study the lamina and calculate the centroid of the lamina from A-A. Round off your answer... show full transcript

Worked Solution & Example Answer:FIGURE 5.1 below shows a shaped lamina - NSC Civil Technology Construction - Question 5 - 2017 - Paper 1

Step 1

Calculate Areas and Centroid X Coordinates

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Answer

For the lamina, divide the shape into two parts: Area 1 (rectangle) and Area 2 (triangle).

Area 1:

Dimensions: 30 mm × 45 mm
Area (A1) = base × height = $30 \, mm imes 45 \, mm = 1350 \, mm^2$

Area 2:

Base = 15 mm, Height = 15 mm
Area (A2) = $\frac{1}{2} \times base \times height = \frac{1}{2} \times 15 \times 15 = 112.5 \, mm^2$

Centroid X Coordinates:

Centroid of Area 1 (X1): X1 = $\frac{30}{2} = 15 \, mm$
Centroid of Area 2 (X2): X2 = $15 + \frac{15}{2} = 22.5 \, mm$

Step 2

Determine the Total Area and Centroid Y Coordinate

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Answer

Total Area (A):

Total area = A1 + A2 = $1350 + 112.5 = 1462.5 \, mm^2$

Centroid Y Coordinates:

Centroid of Area 1 (Y1): Y1 = $\frac{45}{2} = 22.5 \, mm$
Centroid of Area 2 (Y2): Y2 = $45 + \frac{15}{3} = 50 \, mm$

Step 3

Final Calculation for Centroid

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Answer

Now compute the overall centroid (X, Y):

Total Moments about X-axis: $\text{Total Moment X} = A1 \times Y1 + A2 \times Y2 = 1350 \times 22.5 + 112.5 \times 50$ $= 30375 + 5625 = 36000 \, mm^3$
Total Moment about Y-axis: $\text{Total Moment Y} = A1 \times X1 + A2 \times X2 = 1350 \times 15 + 112.5 \times 22.5$ $= 20250 + 2531.25 = 22781.25 \, mm^3$
Centroid coordinates:
$\bar{X} = \frac{Total Moment Y}{A} = \frac{22781.25}{1462.5} \approx 15.56 \, mm$
$\bar{Y} = \frac{Total Moment X}{A} = \frac{36000}{1462.5} \approx 24.59 \, mm$

Final centroid coordinates: (15.56, 24.59) rounded to (15.56, 24.59).

FIGURE 5.1 below shows a shaped lamina - NSC Civil Technology Construction - Question 5 - 2017 - Paper 1

Worked Solution & Example Answer:FIGURE 5.1 below shows a shaped lamina - NSC Civil Technology Construction - Question 5 - 2017 - Paper 1

Calculate Areas and Centroid X Coordinates

Determine the Total Area and Centroid Y Coordinate

Final Calculation for Centroid

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