Question

A circular pizza of radius \(R\) has a circular piece of radius \(R / 4\) removed from one side, as shown in the figure. Where is the center of mass of the pizza with the hole in it?

Short answer

Answer: The center of mass of the pizza with the hole in it is at \(\boxed{\left(-\frac{R}{4},0\right)}\).

Step-by-step solution

Find the area of the entire pizza and the removed piece

First, we need to find the area of the entire pizza and the removed piece. The area of a circle is given by the formula \(A = \pi r^2\), where A is the area and r is the radius. The area of the entire pizza is: \(A_{pizza} = \pi R^2\) The area of the removed piece is: \(A_{removed} = \pi (\frac{R}{4})^2\)

Find the mass of the entire pizza and the removed piece

Let's assume the pizza has a uniform mass distribution, and its mass M is proportional to its area. Therefore, we can find the mass of the entire pizza and the removed piece. Mass of the entire pizza: \(M_{pizza} = kA_{pizza} = k\pi R^2\) (where k is the proportionality constant) Mass of the removed piece: \(M_{removed} = kA_{removed} = k\pi (\frac{R}{4})^2\)

Find the mass of the remaining pizza

To find the mass of the remaining pizza, we'll subtract the mass of the removed piece from the mass of the entire pizza. Mass of the remaining pizza: \(M_{remaining} = M_{pizza} - M_{removed} = k\pi R^2 - k\pi (\frac{R}{4})^2\)

Find the center of mass of the entire pizza and the removed piece

The center of mass of a uniform circle is at its geometric center. The center of mass of the entire pizza is at \((0,0)\), while the center of mass of the removed piece will be on the x-axis, at a distance of \(\frac{3R}{4}\) from the geometric center; that is, at the point \((\frac{3R}{4},0)\).

Find the center of mass of the remaining pizza

To find the center of mass of the remaining pizza, we can use the following formula: $$x_{remaining} = \frac{\sum m_{i}x_{i}}{\sum m_{i}}$$ $$x_{remaining} = \frac{M_{pizza}x_{pizza} - M_{removed}x_{removed}}{M_{remaining}}$$ Plugging in the values: $$x_{remaining} = \frac{(k\pi R^2)(0) - (k\pi (\frac{R}{4})^2)(\frac{3R}{4})}{k\pi R^2 - k\pi (\frac{R}{4})^2}$$ This simplifies to: $$x_{remaining} = \frac{-3R(\frac{1}{16})}{1 - \frac{1}{16}}$$ Solving for \(x_{remaining}:\) $$x_{remaining} = -\frac{3R}{12}$$ Since the y-axis remains unchanged, the center of mass of the remaining pizza will be at the point: $$\left(-\frac{3R}{12},0\right)$$ So, the center of mass of the pizza with the hole in it is at \(\boxed{\left(-\frac{R}{4},0\right)}\).

Key concepts

Physics Problem Solving

Physics problem solving is a systematic approach to understanding and answering questions about the natural world. It involves applying principles of physics to analyze a problem, breaking it down into manageable parts, and using relevant formulas to find a solution.

For example, in the textbook problem about finding the center of mass of a circular pizza with a piece removed, the problem is deconstructed into steps. Each step focuses on determining a particular physical quantity, like area, mass, and center of mass. The key to solving such problems is recognizing that the physical principles at play—like the relationship between mass and area for objects with uniform mass distribution—are consistent and predictable. Students must engage with each part of the problem methodically, ensuring they understand how to employ the physics concepts they’ve learned to reach a logical solution.

Uniform Circular Object

In problems involving a uniform circular object, such as a pizza or a disc, the assumption is often made that the mass is evenly distributed across the object. This simplifies calculations because it means that any slice of the object will have the same mass per unit area as any other slice of equal size.

For objects with uniform mass distribution, the center of mass coincides with the geometric center. When a portion of a uniform circular object is removed, as with the pizza in our example, determining the new center of mass requires considering the mass and position of the removed section in relation to the whole. This balance of mass plays a crucial role in various physics applications, such as the equilibrium of rotating systems, and gives students a tangible understanding of symmetry and distribution in physical objects.

Mass Distribution

Mass distribution concerns how mass is spread out within an object and affects its balance, stability, and motion. It's an essential concept in physics because the distribution of mass determines an object's center of mass, and thus, how it will behave under various forces.

When an object has a uniform mass distribution, it simplifies the process of finding its center of mass. In a situation where mass is removed from an object, like our pizza problem, the center of mass shifts towards the remaining mass. This shift is quantified by considering the position and mass of the removed piece relative to the rest of the object. The way mass is distributed is vital not only in theoretical problems but also in engineering and design, where the balance of a structure or vehicle can affect safety and performance.