Problem 79 - College Physics

Question

A piece of uniform sheet metal measures \(25 \mathrm{~cm}\) by \(25 \mathrm{~cm}\). If a circular piece with a radius of \(5.0 \mathrm{~cm}\) is cut from the center of the sheet, where is the sheet's center of mass now?

Step-by-Step Solution

Answer

The center of mass remains at (12.5 cm, 12.5 cm).

Step 1: Find the Original Center of Mass

Before any cuts, the center of mass of the entire square sheet is at its geometric center. Since the sheet measures \( 25 \text{ cm} \times 25 \text{ cm} \), the original center of mass is at \((12.5, 12.5)\) in cm from any corner.

Step 2: Determine the Masses of Remaining and Removed Pieces

Assuming uniform density, the mass of the original square is proportional to its area: \(625 \text{ cm}^2\). The area of the circular section removed is \(\pi \times (5)^2 = 25\pi \text{ cm}^2\). Thus, the mass remains: \(625 - 25\pi\).

Step 3: Find the Center of Mass of Removed Section

The circular piece is cut symmetrically from the center, so its center of mass is also at \((12.5, 12.5)\) cm.

Step 4: Apply the Center of Mass Formula

The center of mass for composite bodies can be calculated using the formula: \(\bar{x} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2}\) and a similar equation for \(\bar{y}\), where \(m_1\) is mass of the remaining sheet and \(m_2\) is negative of the mass of the removed circle (since it's removed).

Step 5: Calculate the New Center of Mass

Since both the remaining sheet and removed circle have their centers at \((12.5, 12.5)\), the center of mass remains unchanged:\[\bar{x} = \bar{y} = \frac{m_1 \times 12.5 - m_2 \times 12.5}{m_1 - m_2} = 12.5\]This shows that the position of the center of mass does not change.

Key Concepts

Uniform Sheet Metal

Uniform sheet metal refers to a flat piece of metal that has consistent thickness and density throughout. In our exercise, we are dealing with a square sheet of uniform sheet metal measuring 25 cm by 25 cm. The uniformity means that every part of the sheet has the same amount of mass per unit area. This consistent distribution is vital for calculations related to the center of mass.
When calculating the center of mass for such objects, knowing the uniform characteristics allows us to treat the mass distribution as entirely linked to the geometric dimensions.
This simplifies the computation of both areas and masses, key components in determining where the center of mass is located after any modifications, such as cutting out a section.

Geometric Center

The geometric center, often called the centroid in regular shapes, is the point at which a shape's dimensions are balanced. For a square, like the uniform sheet metal in our exercise, the geometric center is at the intersection of its diagonals.
Given that the square measures 25 cm on each side, the geometric center calculates as halfway along both dimensions. This is at \((12.5, 12.5)\), and represents where its center of mass is initially.
Whenever a shape is uniform and regular, its geometric center coincides with its center of mass. Thus, knowing one helps easily find the other in initial problems before any cuts or modifications are made.

Composite Bodies

Composite bodies are objects made up of multiple simple shapes combined together, or in some cases, areas that have been removed, like in our situation.
In solving center of mass problems for composite bodies, the trick lies in understanding that the center of mass is calculated by considering the individual shapes that either remain or have been removed.
The exercise involves removing a circular piece from our square sheet, introducing the concept of negative mass in calculations.

• The remaining square, after cutting out the circle, must be analyzed as a composite.
• The center of mass calculations involve weighting the positions of these bodies against their respective masses.

This requires us to utilize formulas that sum contributions from each part, applying principles of balance and proportional distribution.

Mass Calculation

To effectively solve the problem, precise mass calculation is crucial. Since we're dealing with uniformly distributed mass, the mass of our objects is proportional to their area.
For the original square sheet with an area of \(625\text{ cm}^2\), the total mass is based on this area.
The circular section cut out has an area of \(25\pi\text{ cm}^2\), which needs to be subtracted from the total mass to find the leftover mass after cutting.
The center of mass formula then incorporates these masses to find the new center of mass:

• Use the center of mass equation for composite bodies.
• Consider the remaining mass as positive and the removed mass as negative.
• Since all mass was originally at the center, cutting away mass from the center does not shift the center of mass.

Thus, the center of mass position stays put, emphasizing the importance of mass distribution over pure geometry in such calculations.