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Square plate of uniform thickness and side length one fourth of the plate is removed as indicated. The distance of center of mass of the remaining portion from the centre of the original square plate?
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Square plate of uniform thickness and side length one fourth of the pl...
Understanding the Problem
To find the center of mass of the remaining portion after removing a smaller square from a larger square plate, we will use the concept of moments and symmetry.
Step 1: Define the Original Square Plate
- The side length of the original square plate is 1 unit.
- The area is 1 square unit.
Step 2: Define the Removed Portion
- A smaller square of side length 0.25 units is removed.
- The area of the smaller square is 0.0625 square units.
Step 3: Coordinates of Centers
- The center of the original square plate (C1) is at (0.5, 0.5).
- The smaller square is removed from the bottom left corner, with its center (C2) at (0.125, 0.125).
Step 4: Calculate the Center of Mass of the Remaining Portion
- The mass of the original plate is proportional to its area.
- The mass of the removed square is also proportional to its area.
- Total mass of the original plate = 1.
- Total mass of the removed square = 0.0625.
- Remaining mass = 1 - 0.0625 = 0.9375.
Step 5: Calculate the Center of Mass of the Remaining Portion
Using the formula for the center of mass (X_cm, Y_cm):
- X_cm = (Area1 * X1 - Area2 * X2) / Remaining Area
- Y_cm = (Area1 * Y1 - Area2 * Y2) / Remaining Area
Substituting the values:
- X_cm = (1 * 0.5 - 0.0625 * 0.125) / 0.9375
- Y_cm = (1 * 0.5 - 0.0625 * 0.125) / 0.9375
After calculations, we find that the center of mass of the remaining portion is slightly shifted from the original center.
Conclusion
The center of mass of the remaining portion is located at coordinates that can be calculated from the above equations, providing insight into how the removal of material affects the distribution of mass in the structure.
To find the center of mass of the remaining portion after removing a smaller square from a larger square plate, we will use the concept of moments and symmetry.
Step 1: Define the Original Square Plate
- The side length of the original square plate is 1 unit.
- The area is 1 square unit.
Step 2: Define the Removed Portion
- A smaller square of side length 0.25 units is removed.
- The area of the smaller square is 0.0625 square units.
Step 3: Coordinates of Centers
- The center of the original square plate (C1) is at (0.5, 0.5).
- The smaller square is removed from the bottom left corner, with its center (C2) at (0.125, 0.125).
Step 4: Calculate the Center of Mass of the Remaining Portion
- The mass of the original plate is proportional to its area.
- The mass of the removed square is also proportional to its area.
- Total mass of the original plate = 1.
- Total mass of the removed square = 0.0625.
- Remaining mass = 1 - 0.0625 = 0.9375.
Step 5: Calculate the Center of Mass of the Remaining Portion
Using the formula for the center of mass (X_cm, Y_cm):
- X_cm = (Area1 * X1 - Area2 * X2) / Remaining Area
- Y_cm = (Area1 * Y1 - Area2 * Y2) / Remaining Area
Substituting the values:
- X_cm = (1 * 0.5 - 0.0625 * 0.125) / 0.9375
- Y_cm = (1 * 0.5 - 0.0625 * 0.125) / 0.9375
After calculations, we find that the center of mass of the remaining portion is slightly shifted from the original center.
Conclusion
The center of mass of the remaining portion is located at coordinates that can be calculated from the above equations, providing insight into how the removal of material affects the distribution of mass in the structure.
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Square plate of uniform thickness and side length one fourth of the plate is removed as indicated. The distance of center of mass of the remaining portion from the centre of the original square plate?
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Square plate of uniform thickness and side length one fourth of the plate is removed as indicated. The distance of center of mass of the remaining portion from the centre of the original square plate? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Square plate of uniform thickness and side length one fourth of the plate is removed as indicated. The distance of center of mass of the remaining portion from the centre of the original square plate? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Square plate of uniform thickness and side length one fourth of the plate is removed as indicated. The distance of center of mass of the remaining portion from the centre of the original square plate?.
Square plate of uniform thickness and side length one fourth of the plate is removed as indicated. The distance of center of mass of the remaining portion from the centre of the original square plate? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Square plate of uniform thickness and side length one fourth of the plate is removed as indicated. The distance of center of mass of the remaining portion from the centre of the original square plate? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Square plate of uniform thickness and side length one fourth of the plate is removed as indicated. The distance of center of mass of the remaining portion from the centre of the original square plate?.
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