Question

Find the center of mass of a rectangular plate of length \(20.0 \mathrm{~cm}\) and width \(10.0 \mathrm{~cm} .\) The mass density varies linearly along the length. At one end, it is \(5.00 \mathrm{~g} / \mathrm{cm}^{2}\); at the other end, it is \(20.0 \mathrm{~g} / \mathrm{cm}^{2}\)

Short answer

Answer: To find the center of mass of the rectangular plate, follow the steps outlined in the solution: 1. Set up a coordinate system with the origin at the left end of the plate and the x-axis along its length. 2. Determine the mass of infinitesimally small rectangles dm by using the linear density function for varying mass density along the length. 3. Integrate dm over the entire length to find the total mass M of the plate. 4. Finally, find the x-coordinate of the center of mass by integrating xdm over the length and dividing by M. The y-coordinate will be half of the width w since the mass density is constant along the width. By evaluating these expressions, you can find the center of mass (x_cm, y_cm) of the rectangular plate.

Step-by-step solution

Set up the coordinate system

Place the origin of the coordinate system (x, y) at the left end (density \(5.00 \mathrm{~g}/\mathrm{cm}^{2}\)) of the plate, with x-axis along its length and y-axis along its width.

Determine the mass of the infinitesimally small rectangles

Let's consider dm as the mass of a small rectangle with length dx at a distance x from the left side. The density at this position is given by the linear density function: \(\rho (x) = \rho_0 + kx\) where \(\rho_0 = 5.00 \mathrm{~g}/\mathrm{cm}^{2}\) is the initial density, k is the rate of change of density, and x is the distance from the left side in cm. To find k, use the density at the other end of the plate. \(\rho(20) = 20.0 \mathrm{~g}/\mathrm{cm}^{2} = 5.00 + 20k\) Solve for k: \(k = \frac{15}{20} \mathrm{g}/\mathrm{cm}^3\) Now, determine dm as follows: \(dm = \rho(x)w dx = (5 + \frac{15x}{20})w dx\), where w is the width of the plate in cm.

Calculate the mass of the entire plate

To find the total mass of the plate, integrate dm over the entire length: \(M = \int_0^{20} dm = \int_0^{20} (5 + \frac{15x}{20})wdx\)

Determine the center of mass

The x-coordinate of the center of mass is given by: \(x_{cm} = \frac{1}{M} \int_0^{20} x dm\) Substitute the expression for dm from Step 2 into this formula: \(x_{cm} = \frac{1}{M} \int_0^{20} x (5 + \frac{15x}{20})wdx\) Now, perform the integration and divide by the total mass M to get the x-coordinate of the center of mass. Finally, since the mass density is constant along the width, the y-coordinate of the center of mass will be at the center of the width, i.e.: \(y_{cm} = \frac{w}{2}\) The center of mass (x_cm, y_cm) can now be found by evaluating these expressions.

Key concepts

Mass Density

The concept of mass density is fundamental to understanding many physical situations, including the calculation of a center of mass. In general, density is a measure of how much mass is contained in a given volume. It is commonly expressed in units such as grams per cubic centimeter (g/cm³) for solids and liquids, or kilograms per cubic meter (kg/m³) for gases.

However, when dealing with two-dimensional objects like the plate in the exercise, we refer to surface mass density, which is mass per unit area, and has units like grams per square centimeter (g/cm²). In other words, it expresses the amount of mass spread over a particular area of the surface. For a plate with uniform density, determining its mass would simply require multiplying the density by the area. But, when the density varies across the length or width of an object, we should approach the problem by considering differential elements, as shown in the provided solution.

Linear Density Function

A linear density function describes how the mass density of an object varies along a single dimension. In the case of the rectangular plate, the mass density varies linearly from one end to the other. This is characterized by a function of the form \( \rho(x) = \rho_0 + kx \) where \( \rho_0 \) is the density at one reference point, and 'k' is a constant that represents the rate of change of the density along the length.

This mathematical model is a powerful tool because it allows us to calculate the mass of infinitesimally small sections of the object by integrating the density function over the object's length. Without this function, it would be quite impossible to compute the center of mass for non-uniform density distributions accurately.

The linear density function implies that the amount of mass in each small section varies in a predictable way, which in turn affects the balance of the object. Therefore, the proper use of this function is crucial for calculating the center of mass, as seen in the provided step-by-step solution.

Coordinate System

The coordinate system is a systematic way of defining the position of points or elements in space. For most problems in physics, especially in calculating the center of mass, a Cartesian coordinate system is used, consisting of an x-axis and a y-axis that are perpendicular to each other.

In our exercise, setting up the coordinate system is the first step toward solving the problem. The origin is placed at one end of the plate, allowing us to define a variable 'x' that represents the distance from this point. The choice of origin is usually made to simplify the calculation, often at a symmetry point or boundary of the object.

The coordinate system is essential because it allows us to express the position of the center of mass in terms of coordinates (x_cm, y_cm). This not only gives us a precise location of the center of mass but also an insightful understanding of how mass is distributed across the object. It is also critical for integrating the mass and the distance, x, from the chosen origin, which ultimately leads us to find the center of mass.