a circular section 8.0cm in diameter is marked out 9f a rectangular plate 20cm ×18cm in dimensions. if the centre of the section marked out lie on the diagonal abd is 7.0cm from the centre of gravity of the rectangular plate, determin the position of the centre of gravity of the remaining plate if the section marked out is eventually cut out.?
To find the position of the center of gravity of the remaining plate, we can break down the problem into smaller steps.
Step 1: Determine the position of the center of the circular section.
Given that the circular section has a diameter of 8.0 cm, the radius is half of the diameter, 8.0 cm / 2 = 4.0 cm. Since the center of the circular section lies on the diagonal, it divides the diagonal into two equal segments. Let's call the distance from the center of the rectangular plate to the center of the circular section "d". According to the problem, d = 7.0 cm.
Step 2: Find the length of the diagonal of the rectangular plate.
Applying the Pythagorean theorem, the diagonal of the rectangular plate, let's call it "D", can be calculated as follows:
D = √(20^2 + 18^2)
Step 3: Calculate the remaining length of the diagonal.
Since the circular section lies on the diagonal, cutting it out will reduce the length of the diagonal by the diameter of the circle (8.0 cm). Therefore, the remaining length of the diagonal after cutting out the circular section is given by:
remaining_diagonal = D - 8.0 cm
Step 4: Determine the position of the center of gravity of the remaining plate.
Since the center of gravity of a rectangle lies at the midpoint of its diagonal, we can find the position of the center of gravity of the remaining plate by setting the distance from the center of gravity of the rectangular plate to the center of the circular section equal to the remaining diagonal divided by 2:
remaining_center_of_gravity = remaining_diagonal / 2
Finally, substitute the values we obtained in Steps 2 and 4 to find the solution.
To determine the position of the center of gravity of the remaining plate, we can follow these steps:
1. Calculate the position of the original center of gravity of the rectangular plate.
Given:
- Dimensions of the rectangular plate: 20cm × 18cm
- Center of gravity of the rectangular plate is 7.0cm from the center of the circular section.
The center of gravity of a rectangular plate is given by the formula:
x_cg = (L/2) - (x/2)
where:
- x_cg is the position of the center of gravity
- L is the length of the plate
- x is the distance from the center of the plate to the center of gravity
Plugging in the values, we have:
x_cg = (20/2) - (7/2)
x_cg = 10 - 3.5
x_cg = 6.5 cm
2. Calculate the new position of the center of gravity after the circular section is cut out.
Given:
- Diameter of the circular section: 8.0 cm
- Radius of the circular section: 8.0 cm / 2 = 4.0 cm
The circular section is cut out, so its diameter will be removed from the rectangular plate. To find the new position of the center of gravity, we need to adjust the original position by considering the circular section’s dimensions.
Since the circular section is centered on the diagonal, the x-coordinate of the center of gravity will remain the same (6.5 cm). However, the y-coordinate needs to be adjusted by taking into account the radius of the circular section.
The y-coordinate adjustment is given by the formula:
Δy = (r/L) * (L/2)
where:
- Δy is the adjustment in the y-coordinate
- r is the radius of the circular section
- L is the length of the plate
Plugging in the values, we have:
Δy = (4.0/18) * (20/2)
Δy = (2/9) * 10
Δy = 20/9
Δy ≈ 2.22 cm
Therefore, the new position of the center of gravity of the remaining plate is:
x = 6.5 cm
y = original y-coordinate - Δy
y = 7.0 cm - 2.22 cm
y ≈ 4.78 cm
Hence, the position of the center of gravity of the remaining plate is approximately (6.5 cm, 4.78 cm).