---
title: "A small vehicle of mass \\(m\\) travels at a constant speed \\(v\\) in a horizontal circular path of radius \\(R\\) along the frictionless inner surface of a symmetric bowl. The profile height \\(y\\) of the surface as a function of radial distance \\(r\\) from the vertical central axis is given by \\(y(r) = \\beta r^3\\), where \\(\\beta\\) is a positive constant. Which of the following expressions represents the speed \\(v\\) required for the vehicle to maintain its circular path at radius \\(R\\) without sliding up or down the surface?"
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url: "https://nerd-notes.com/ubq/120595/"
date_modified: "2026-08-23T04:41:48+00:00"
---

# A small vehicle of mass \(m\) travels at a constant speed \(v\) in a horizontal circular path of radius \(R\) along the frictionless inner surface of a symmetric bowl. The profile height \(y\) of the surface as a function of radial distance \(r\) from the vertical central axis is given by \(y(r) = \beta r^3\), where \(\beta\) is a positive constant. Which of the following expressions represents the speed \(v\) required for the vehicle to maintain its circular path at radius \(R\) without sliding up or down the surface?

A small vehicle of mass \(m\) travels at a constant speed \(v\) in a horizontal circular path of radius \(R\) along the frictionless inner surface of a symmetric bowl. The profile height \(y\) of the surface as a function of radial distance \(r\) from the vertical central axis is given by \(y(r) = \beta r^3\), where \(\beta\) is a positive constant. Which of the following expressions represents the speed \(v\) required for the vehicle to maintain its circular path at radius \(R\) without sliding up or down the surface?

![A grayscale cross-sectional diagram of a smooth, upward-opening curved bowl symmetric about a vertical dashed centerline labeled as the y-axis. The base of the bowl rests at the origin. A small solid rectangular block representing the vehicle is positioned on the right inner surface of the bowl at a horizontal distance R from the vertical centerline. A horizontal dashed circle with radius arrow labeled R indicates the horizontal plane of motion. At the point of contact beneath the block, a straight solid tangent line forms an angle \theta with a horizontal dashed reference line. A normal force vector arrow \vec{F}_N points perpendicularly inward from the surface through the block, and a downward gravitational force vector arrow m\vec{g} points vertically downward from the block. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787460108-7iQA9E.jpg)

- **A.** \(\sqrt{\dfrac{g}{3\beta R}}\)
- **B.** \(\sqrt{\dfrac{\beta g R^3}{3}}\)
- **C.** \(\sqrt{\beta g R^3}\)
- **D.** \(\sqrt{3\beta g R^3}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/120595/*