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: The center of mass of the rectangular plate is at 10 cm from the left end along the length (x-axis).

Step-by-step solution

Define variables and given information

Length of rectangular plate, L = \(20.0 \thinspace cm\) Width of rectangle, W = \(10.0 \thinspace cm\) Mass density at one end, \(\rho_1 = 5.00 \thinspace g/cm^2\) Mass density at the other end, \(\rho_2 = 20.0 \thinspace g/cm^2\) Let x-axis be along the length and y-axis along the width.

Calculate the total mass of the rectangular plate

To find the total mass, we must find the mass per unit length along the x-axis and then integrate along the length. Let \(\rho(x)\) be the mass density function along the length. Since the mass density varies linearly, we can write: \(\rho(x) = \rho_1 + (\rho_2 - \rho_1) \frac{x}{L}\) = \(5 + (20 - 5) \frac{x}{20} = 5 + \frac{3}{4}x\) Now, for mass per unit length, \(dm = \rho(x) \thinspace dA = \rho(x) \thinspace W \thinspace dx\)

Integrate the mass per unit length to find the total mass

Integrate \(dm\) to find the total mass, M: \(M = \int_{0}^{L} dm = \int_{0}^{20} (5 + \frac{3}{4} x) (10) dx = 10 \int_{0}^{20} (5 + \frac{3}{4} x) dx\) \(M = 10[5x + \frac{3}{8} x^2]_0^{20} = 10 (100 + \frac{3}{8} \cdot 400) = 2200 \thinspace g\)

Find the center of mass along the x-axis

To find the center of mass along the x-axis, we use the formula: \(x_{cm} = \frac{1}{M} \int_{0}^{L} x \thinspace dm\) Integrating this expression we get: \(x_{cm} = \frac{1}{2200} \int_{0}^{20} x(10)(5 + \frac{3}{4} x) dx = \frac{1}{220} \int_{0}^{20} (50x + \frac{15}{4} x^2) dx\) \(x_{cm} = \frac{1}{220} [25x^2 + \frac{15}{12} x^3]_0^{20} = \frac{1}{220} (10000 + 10000) = \frac{20000}{220}\) Simplifying the expression we obtain: \(x_{cm} = 10 \thinspace cm\) The center of mass of the rectangular plate is at 10 cm from the left end along the length(x-axis).

Key concepts

Mass Density

Mass density, often denoted by the symbol \( \rho \), is a measure of how 'crowded' a substance is with mass. Think of it as how much stuff is packed into a specific space. For a rectangular plate, we usually talk about surface mass density, which is the mass per unit area. It's a bit different from the mass density of a 3D object, which is mass per unit volume.

In the case of our rectangular plate exercise, mass density doesn’t stay constant; it changes linearly from one end to the other. This linear change is an example of a non-uniform mass distribution. You can picture this like a gradient, where one end is lighter (less mass) and the other is heavier (more mass), much like a color fading from light to dark across the surface.

Integration in Physics

In physics, integration is a mathematical tool that allows us to add up an infinite number of infinitesimally small quantities to determine a whole. It's perfect for dealing with variables that change continuously, like the mass density in our rectangular plate example.

When we calculate the total mass 'M', we integrate the small masses (\( dm \)s) across the entire surface. Each \( dm \) represents a tiny slice of the entire plate—just like if you sliced a gradated block of cheese from white to yellow and weighed each thin slice. For the calculated \( dm \)s, the width 'W' is constant, but mass density \( \rho(x) \) varies with length 'x', which yields a function we integrate over the entire length 'L' of the plate.

Rectangular Plate Mass Distribution

Dealing with the mass distribution of a rectangular plate, especially when the density is not uniform, requires breaking down the plate into differential elements where the density can be considered uniform. In our case, each differential strip runs parallel to the width and is infinitesimally thin, with a density that's a function of 'x'.

The solution method, which involves integration across these strips, essentially sums up the contributions from all the strips to find the total mass. Due to the linear variation in mass density, the center of mass won’t be smack dab in the middle of the plate but rather will shift towards the heavier side. By calculating the center of mass using integration, we take into account all point masses across the distribution, essentially finding the 'balance point' of the rectangular plate.