---
title: "A particle moves in the $xy$-plane such that its position vector as a function of time $t$ is given by $\\vec{r}(t) = (b t \\cos(\\omega t))\\hat{i} + (b t \\sin(\\omega t))\\hat{j}$, where $b$ and $\\omega$ are positive constants. What is the ratio of the magnitude of the $y$-component of the acceleration to the magnitude of the $x$-component of the acceleration, $\\dfrac{|a_y|}{|a_x|}$, at time $t = \\dfrac{\\pi}{2\\omega}$?"
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url: "https://nerd-notes.com/ubq/117624/"
date_modified: "2026-08-04T07:49:21+00:00"
---

# A particle moves in the $xy$-plane such that its position vector as a function of time $t$ is given by $\vec{r}(t) = (b t \cos(\omega t))\hat{i} + (b t \sin(\omega t))\hat{j}$, where $b$ and $\omega$ are positive constants. What is the ratio of the magnitude of the $y$-component of the acceleration to the magnitude of the $x$-component of the acceleration, $\dfrac{|a_y|}{|a_x|}$, at time $t = \dfrac{\pi}{2\omega}$?

A particle moves in the $xy$-plane such that its position vector as a function of time $t$ is given by $\vec{r}(t) = (b t \cos(\omega t))\hat{i} + (b t \sin(\omega t))\hat{j}$, where $b$ and $\omega$ are positive constants. What is the ratio of the magnitude of the $y$-component of the acceleration to the magnitude of the $x$-component of the acceleration, $\dfrac{|a_y|}{|a_x|}$, at time $t = \dfrac{\pi}{2\omega}$?

![An $xy$-coordinate plane with a horizontal $x$-axis labeled $x$ and a vertical $y$-axis labeled $y$ intersecting at an origin labeled $O$. A smooth curve depicting an outward-expanding counterclockwise spiral trajectory starts at $O$, passes through the first quadrant, and crosses the positive $y$-axis at point $P$. An arrow labeled $\vec{r}$ points from $O$ to point $P$ along the $y$-axis. Black arrowheads at the ends of the axes indicate positive directions. A small unit vector arrow labeled $\hat{i}$ points rightward along the positive $x$-axis from $O$, and a small unit vector arrow labeled $\hat{j}$ points upward along the positive $y$-axis from $O$. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785829761-UkftkL.jpg)

- **A.** \(\dfrac{\pi}{8}\)
- **B.** \(\dfrac{\pi}{4}\)
- **C.** \(\dfrac{\pi}{2}\)
- **D.** \(\pi\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117624/*