Question

Explain why or why not ,Determine whether the following statements are true and give an explanation or counterexample. a. A thin plate of constant density that is symmetric about the \(x\) -axis has a center of mass with an \(x\) -coordinate of zero. b. A thin plate of constant density that is symmetric about both the \(x\) -axis and the \(y\) -axis has its center of mass at the origin. c. The center of mass of a thin plate must lie on the plate. d. The center of mass of a connected solid region (all in one piece) must lie within the region.

Short answer

Explain your reasoning. Answer: Yes, the center of mass of a thin plate with constant density that is symmetric about both the x-axis and the y-axis is always located at the origin. This is because the symmetry about both axes results in the equal distribution of mass along these axes, causing the forces on all sides to balance each other out. Consequently, the center of mass will have an x-coordinate of zero and a y-coordinate of zero, placing it at the origin with the coordinates (0,0).

Step-by-step solution

a. A thin plate of constant density that is symmetric about the x-axis has a center of mass with an x-coordinate of zero.

This statement is true. Since the plate is symmetric about the x-axis, there is an equal amount of mass on both sides of the x-axis. As a result, the forces on both sides of the x-axis will balance each other out, causing the center of mass to lie along the x-axis with an x-coordinate of zero.

b. A thin plate of constant density that is symmetric about both the x-axis and the y-axis has its center of mass at the origin.

This statement is also true. Similar to part (a), the symmetry about the x-axis will cause the plate to have a center of mass with an x-coordinate of zero. Additionally, since the plate is symmetric about the y-axis, there will be an equal amount of mass above and below the y-axis, causing the forces on both the top and bottom halves to balance each other out. This will result in the center of mass having a y-coordinate of zero as well. Therefore, the center of mass will be at the origin with coordinates (0,0).

c. The center of mass of a thin plate must lie on the plate.

This statement is false. A counterexample can be a semi-circular plate. The center of mass of a semi-circular plate will lie at a point halfway along its radius, inside of the semi-circle, even though that point is not part of the plate itself.

d. The center of mass of a connected solid region (all in one piece) must lie within the region.

This statement is true. For any connected solid region, the center of mass must lie within the region as it represents the average position of all the particles in the system. The center of mass is the point about which the torque due to the gravitational force is zero, meaning that the object would balance perfectly if it were to pivot about that point. Since the mass distribution in a connected region is continuous and not fragmented, the center of mass must be inside the region for the forces to balance out properly.

Key concepts

Symmetry in Calculus

Symmetry is a powerful concept in calculus that greatly simplifies the calculation of various properties, such as the center of mass. It refers to the balance and proportional correspondence of parts on opposite sides of a dividing line or about a center or axis. In problems involving geometric objects, if an object is symmetrical about an axis, any line integral or integral of a function over that object can often be computed by considering just one portion of the object and then multiplying by a factor that accounts for the symmetry.

For example, when a thin plate of uniform density has symmetry about the x-axis, intuitively, the material is distributed equally on either side of the x-axis. This equal distribution means the x-coordinate of the center of mass must be zero. In the case where the plate is symmetric about both the x-axis and the y-axis, the center of mass must lie at the origin, since both the x and y coordinates of the center of mass must be zero, reflecting the balance in both dimensions.

Application of Symmetry in Calculus

Symmetry is not only useful in locating the center of mass but is also beneficial in simplifying complex integrals. When recognizing symmetry, we can limit the integration to just half (or another fraction) of the object, thus reducing the computational effort. It is one of the strategic tools that highlight the interconnectedness of geometry and calculus.

Density in Calculus

In calculus, density typically refers to mass per unit volume or mass per unit area in the context of a two-dimensional figure like a thin plate. Density can be a function that varies within a substance, or it can be constant across the object.

Considering a constant density simplifies the calculation of the center of mass. It ensures that the mass distribution is uniform across the object, which aids in predicting the behavior of the object under forces such as gravity. In our exercise, the assumption of a constant density allows us to focus solely on the geometry of the plates to determine their centers of mass.

Uniform vs. Variable Density

In real-world applications, density may not be consistent. For instance, the density of an object might change with depth or due to material properties. Calculating the mass, and by extension properties like the center of mass, becomes more complex with variable density. In such cases, integral calculus comes into play, requiring the use of variable-density functions and triple integrals for three-dimensional objects to calculate these properties accurately.

Counterexamples in Calculus

Counterexamples are a fundamental aspect in calculus and mathematics in general. A counterexample demonstrates that a statement is not always true. It is a specific case or instance that refutes a conjecture or a universal assertion. When we claim that something is false in mathematics, providing a counterexample is a powerful way of proving our point.

In our exercise, for the statement that 'the center of mass of a thin plate must lie on the plate,' a counterexample is offered to refute it: a semi-circular plate. This is because the center of mass does not necessarily have to be a point that is part of the physical plate, especially if the plate is not a complete geometric figure.

Importance of Counterexamples

Understanding the concept of counterexamples teaches students to embrace the complexity of calculus concepts and not to take assertions at face value without consideration of special cases or exceptions. It helps in grasping the confines within which a mathematical statement can be considered true.

Connected Solid Regions in Calculus

Connected solid regions pertain to bodies that are in one piece, with no separations or gaps, and whose mass is distributed continuously. In calculus, when dealing with these regions, various properties such as volume, surface area, and the center of mass can be calculated using integration. Since these objects are continuous and without fragmentation, there's an inherent assumption that physical properties like density are also continuous within them.

The center of mass for a connected solid region is a point that is integral in understanding how the object would balance under the force of gravity, or how it might rotate about an axis. This is evident in our exercise, which states that the center of mass of such a region must lie within that region. The intuitive understanding is that since the substance is continuous and intact, the average position of the mass distribution - the center of mass - must also reside within that distribution.

Practical Relevancy in Calculus

A solid comprehension of connected solid regions aids in the movement from theoretical calculus concepts to practical applications such as engineering and physics, where it's crucial to predict the behavior of materials and structures under various forces and conditions.