Problem 544 - JEE Physics

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. \(\\{\mathrm{A}\\}-7 \mathrm{~cm} \quad\\{\mathrm{~B}\\}+9 \mathrm{~cm} \quad\\{\mathrm{C}\\}-9 \mathrm{~cm} \quad\\{\mathrm{D}\\}+7 \mathrm{~cm}\)

Step-by-Step Solution

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 1: 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.

Step 2: 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.

Step 3: 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 \((24.5,0)\).

Step 4: 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.

Key Concepts

Uniform Circular Plate

A **uniform circular plate** is a disk where every part is evenly distributed in terms of mass and thickness. This means that any segment of the plate has the same mass density as any other segment of the same size. Because of this uniformity, it makes it easier to calculate properties like center of mass and moment of inertia.
When dealing with a uniform circular plate, its center of mass is at its geometric center, assuming no portions have been removed or added. This simplifies calculations significantly as we can assume symmetry along the center line of the plate. For example, for a full circular plate with radius \( R \), its center of mass naturally lies at the origin \((0,0)\) if placed on a coordinate system.

Circular Area Subtraction

**Circular area subtraction** is a vital concept when dealing with composite shapes where a section has been removed. In this exercise, a smaller circular section is cut out from a larger uniform circular plate, affecting its overall balance and center of mass.
To subtract a circular area, we calculate the area of the section that is being removed. We do this by using the formula for the area of a circle, \(A = \pi r^2\), where \( r \) is the radius of the circle. By finding the area of the removed part, we can adjust the calculations for the new center of mass of the remaining part of the plate.
Once the area is subtracted, the plate is no longer symmetrical around its original center, which means the center of mass will shift. Understanding this shift is key to solving problems related to center of mass in systems with removed sections.

Distance Between Centers

The **distance between centers** of two circular areas is a crucial measurement necessary for determining how a removal or addition will affect the system's center of mass. In this problem, it refers to the distance from the center of mass of the entire original plate to the center of mass of the section that's been taken out.
This is calculated by considering the midpoint between their boundaries when one circle is placed at the origin. With one circle defined by radius \( R1 \) and another by \( R2 \), the distance between their centers \( d \) is determined by the formula:

• \( d = \frac{R1 + R2}{2} \)

Using this distance is essential to locate the exact position of a cut-out and subsequently recalibrate or find the new center of mass for the remaining object.

Center of Mass Formula

The **center of mass formula** is the mathematical tool we use to find the point at which the mass of a composite shape seems to be concentrated. In the context of a uniform circular plate with a portion removed, this formula helps us find where the remaining mass balances out.
The formula often looks like this in two-dimensional space:

• \( M \bar{x} = \sum (m_i x_i) \)

Where:

• \( M \) is the total mass.
• \( \bar{x} \) is the center of mass on the x-axis.
• \( m_i \) is the mass of each part, and \( x_i \) is the distance from a reference point.

For our problem, using the center of mass formula, we subtract out the portion (cut-out) by treating its mass as negative mass:

• \( A1 imes 0 - A2 imes d = \Delta x \times (A1 - A2) \)

Solving for \( \Delta x \) gives the shift in the center of mass, indicating how far to move from the initial center to find the balance point of the remaining mass.