Question

A circular plate of uniform thickness has a diameter of \(56 \mathrm{~cm}\). A circular portion of diameter \(42 \mathrm{~cm} .\) is removed from tve \(\mathrm{x}\) edge of the plate. Find the position of centre of mass of the remaining portion with respect to centre of mass of whole plate. $\\{\mathrm{A}\\}-7 \mathrm{~cm} \quad\\{\mathrm{~B}\\}+9 \mathrm{~cm} \quad\\{\mathrm{C}\\}-9 \mathrm{~cm} \quad\\{\mathrm{D}\\}+7 \mathrm{~cm}$

Short answer

The position of the center of mass of the remaining portion with respect to the center of mass of the whole plate is approximately \(-9\, cm\), which corresponds to the answer option C.

Step-by-step solution

Understanding the given information and definitions

Diameter of the whole circular plate = \(56 cm\) Diameter of the cut-out circular portion = \(42 cm\) We can now calculate the radii of each circle and their respective areas, which are necessary to later find their center of mass positions.

Find the radii and areas of the circles

Radius of the whole circular plate (\(R1\)) = \(\frac{56}{2}=28 cm\) Radius of the cut-out circular portion (\(R2\)) = \(\frac{42}{2}=21 cm\) Area of the whole circular plate (\(A1\)) = \(π R1^2 = π (28)^2 = 2464π\, cm^2\) Area of the cut-out circular portion (\(A2\)) = \(π R2^2 = π (21)^2 = 1323π\, cm^2\) Since the plate has a uniform thickness, we do not need to consider the height of the plate.

Find the center of mass positions

Let the position of the center of mass of the whole plate be at the origin, i.e., \((0,0)\) It is given that the cut-out portion has been removed from the edge, which means the distance between the centers of the two circles (the center of the larger plate and the center of the cut-out portion) is equal to half the sum of their radii: Distance between centers (\(d\)) = \( \frac{R1 + R2}{2} = \frac{28+21}{2} = 24.5 cm\) Thus, the coordinates of the center of mass of the cut-out portion are \((24.5,0)\).

Find the position of the center of mass of the remaining portion with respect to the center of mass of the whole plate

Let the position of the center of mass of the remaining portion with respect to the center of mass of the whole plate be \(\Delta x\). Using the center of mass formula, we can write: \(A1(0) - A2(24.5) = (\Delta x)(A1 - A2)\) Here, \(A1\) and \(A2\) are the areas of the whole plate and the cut-out circular portion, respectively. Now, substitute the known values of the areas and distance: \(2464π(0) - 1323π(24.5) = (\Delta x)(2464π - 1323π)\) Solving for \(\Delta x\), we get: \(\Delta x = \frac{-1323π(24.5)}{1141π} ≈ -9\, cm\) Therefore, the position of the center of mass of the remaining portion with respect to the center of mass of the whole plate is approximately \(-9\, cm\), which corresponds to the answer option C.