---
title: "A thin rectangular plate of length \\(L\\) and width \\(W\\) lies in the \\(xy\\)-plane bounded by \\(x = 0\\) to \\(x = L\\) and \\(y = 0\\) to \\(y = W\\). The surface mass density of the plate is non-uniform and given by \\(\\sigma(x) = \\sigma_0 \\left(\\dfrac{x}{L}\\right)^2\\), where \\(\\sigma_0\\) is a constant. What is the \\(x\\)-coordinate of the center of mass of the plate?"
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url: "https://nerd-notes.com/ubq/117593/"
date_modified: "2026-08-04T07:49:14+00:00"
---

# A thin rectangular plate of length \(L\) and width \(W\) lies in the \(xy\)-plane bounded by \(x = 0\) to \(x = L\) and \(y = 0\) to \(y = W\). The surface mass density of the plate is non-uniform and given by \(\sigma(x) = \sigma_0 \left(\dfrac{x}{L}\right)^2\), where \(\sigma_0\) is a constant. What is the \(x\)-coordinate of the center of mass of the plate?

A thin rectangular plate of length \(L\) and width \(W\) lies in the \(xy\)-plane bounded by \(x = 0\) to \(x = L\) and \(y = 0\) to \(y = W\). The surface mass density of the plate is non-uniform and given by \(\sigma(x) = \sigma_0 \left(\dfrac{x}{L}\right)^2\), where \(\sigma_0\) is a constant. What is the \(x\)-coordinate of the center of mass of the plate?

![A rectangular plate in the xy-plane with its lower-left corner at the origin. The plate extends horizontally along the x-axis from 0 to L and vertically along the y-axis from 0 to W. The density of shading increases smoothly along the x-axis from left to right, being white at x = 0 and dark gray at x = L. A horizontal arrow along the bottom edge is labeled x, reaching x = L, and a vertical arrow along the left edge is labeled y, reaching y = W. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785829754-OPz3do.jpg)

- **A.** \(\dfrac{1}{2}L\)
- **B.** \(\dfrac{2}{3}L\)
- **C.** \(\dfrac{3}{4}L\)
- **D.** \(\dfrac{4}{5}L\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117593/*