A thin plate with uniform thickness is made of homogeneous material. The dimension is as shown (unit in mm). The plate is symmetrical about Axis xx. Calculate the location of centre of mass.

Question

BlackTom AI · Accepted Answer

Let me restate the problem and options, then break down the reasoning for each choice.
Question and options:
- A thin plate with uniform thickness, outer dimensions 50 mm (width) by 40 mm (height). A rectangular cutout (hole) of width 25 mm and height 20 mm is removed from the right side, leaving a symmetric C-shaped plate. The centroid of the whole shape is sought along the x-direction from the left edge. Options for the location of the center of mass (from the left edge):
  1) 20.83 mm
  2) 18.66 mm
  3) 15 mm
  4) 20 mm

Option 1: 20.83 mm from the left edge
- This result comes from treating the shape as a large outer rectangle minus a rectangular hole (composite area method).
- Outer rectangle: width 50 mm, height 40 mm. Area A_out = 50 × 40 = 2000 mm². Its centroid is at x_out = 25 mm (half of width) from the left edge, and y_out = 20 mm (half of height).
- Hole (cutout): width 25 mm, height 20 mm. Area A_hole = 25 × 20 = 500 mm². Its centroid is at x_h = left edge of hole + 25/2. The hole is placed flush with the right edge, so its left edge is at x = 25 mm, giving x_h = 25 + 12.5 = 37.5 mm. Its y-coordinate is 20 mm, same as the plate’s mid-height.
- Composite centroid along x is given by x_cm = (A_out·x_out − A_hole·x_h) / (A_out − A_hole).
  Compute: A_out·x_out = 2000 × 25 = 50,000; A_hole·x_h = 500 × 37.5 = 18,750; net moment = 50,000 − 18,750 = 31,250; net area = 2000 − 500 = 1500.
  Therefore x_cm = 31,250 / 1,500 = 20.833... mm, i.e., 20.83 mm.
- This aligns with a symmetric vertical placement (mid-height) and hole flush on the right, yielding a centroid shifted left of the center (25 mm). The calculation is consistent with the numbers above.

Option 2: 18.66 mm from the left edge
- If you got 18.66 mm, you likely misidentified the hole position or misapplied the areas. For instance, assuming a different hole position (not flush with the right edge) or misplacing the hole centroid would change the moment balance. With the given geometry (hole flush to the right edge, width 25, height 20), the centroid result is 20.83 mm, not 18.66 mm.

Option 3: 15 mm from the left edge
- A centroid at 15 mm would imply a much larger leftward shift, which would require a hole location or a different hole size that produces a greater leftward moment than the actual geometry allows. With the actual numbers (A_out = 2000, A_hole = 500, x_out = 25, x_h = 37.5), the centroid cannot be as far left as 15 mm.

Option 4: 20 mm from the left edge
- This is close to the correct value but still inaccurate for the precise given geometry. The exact computation with the stated dimensions yields 20.833... mm, which rounds to 20.83 mm rather than exactly 20.0 mm. The discrepancy arises from the same composite-area calculation described for Option 1.

Summary: The correct approach is to model the plate as an outer rectangle minus the rectangular cutout, compute the areas and centroids of both, and apply the composite centroid formula. Using A_out = 2000 mm² with x_out = 25 mm and A_hole = 500 mm² with x_h = 37.5 mm yields x_cm = 20.833... mm from the left edge, i.e., 20.83 mm. The other options can be ruled out by applying the same method and noting the exact geometry (hole flush with the right edge, 25×20 hole) which fixes the centroid at 20.83 mm.}

A thin plate with uniform thickness is made of homogeneous material. The dimension is as shown (unit in mm). The plate is symmetrical about Axis xx. Calculate the location of centre of mass. 单项选择题

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