A thin rectangular plate of uniform areal density σ = 2.79 kg/m2 has length of 37.0 cm and width of 23.0 cm. The lower left hand corner is located at the origin, (x,y)= (0,0) and the length is along the x-axis.
(a)There is a circular hole of radius 8.00 cm with center at (x,y) = (12.50,9.50) cm in the plate. Calculate the mass of plate.
(b)Calculate the distance of the plate's CM from the origin
I'm interested in part b! thank you!
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To calculate the distance of the plate's center of mass (CM) from the origin, we need to find the x-coordinate and y-coordinate of the CM separately, and then calculate the overall distance using the Pythagorean theorem.
Let's start with finding the x-coordinate of the CM.
The plate can be divided into two parts: the rectangle and the circular hole. Since they have different densities, we need to consider each part separately.
For the rectangular part:
The area of the rectangle is given by A_rectangle = length * width = (37.0 cm) * (23.0 cm) = 851.0 cm².
The mass of the rectangular part is given by m_rectangle = σ * A_rectangle = (2.79 kg/m²) * (851.0 cm²) / (10,000 cm²/m²) = 0.236 meters.
The x-coordinate of the center of mass for the rectangular part can be determined by considering that it is at the center of the rectangle, which is at x = length / 2 = (37.0 cm) / 2 = 18.5 cm.
For the circular hole part:
The area of the circular hole is given by A_hole = π * (radius_hole)² = π * (8.00 cm)² = 201.06 cm².
The mass of the circular hole part is given by m_hole = σ * A_hole = (2.79 kg/m²) * (201.06 cm²) / (10,000 cm²/m²) = 0.056 meters.
The x-coordinate of the center of mass for the circular hole part can be determined by considering that it is at the x-coordinate of the center of the circle, which is at x = 12.50 cm.
Now, we can calculate the overall x-coordinate of the center of mass (CM_x) by considering the masses and positions of the rectangular and circular hole parts:
CM_x = (m_rectangle * x_rectangle + m_hole * x_hole) / (m_rectangle + m_hole)
= (0.236 kg * 18.5 cm + 0.056 kg * 12.50 cm) / (0.236 kg + 0.056 kg)
= 4.89 cm
Next, let's find the y-coordinate of the CM.
The y-coordinate of the center of mass for both the rectangular and circular hole parts can be determined by considering that they are at the center of the plate, which is at y = width / 2 = 23.0 cm / 2 = 11.5 cm.
Finally, we can calculate the overall distance (d) of the plate's center of mass from the origin using the Pythagorean theorem:
d = √(CM_x² + CM_y²)
= √((4.89 cm)² + (11.5 cm)²)
= √(23.90 cm² + 132.25 cm²)
= √(156.15 cm²)
≈ 12.50 cm
Therefore, the distance of the plate's center of mass from the origin is approximately 12.50 cm.