---
title: "A cart on a horizontal track accelerates to the right with a constant acceleration of magnitude \\(a\\). A small block of mass \\(m\\) is held against the front face of the cart, which is inclined at an angle \\(\\theta\\) above the horizontal (\\(0 < \\theta \\le 90^\\circ\\)). The coefficient of static friction between the block and the inclined face is \\(\\mu_s\\), such that the minimum acceleration required to prevent the block from slipping down the face is given by: \\[ a_{\\text{min}} = g\\left(\\dfrac{\\sin\\theta – \\mu_s \\cos\\theta}{\\cos\\theta + \\mu_s \\sin\\theta}\\right) \\] Which of the following correctly describes the limiting behavior of \\(a_{\\text{min}}\\) as the face becomes vertical (\\(\\theta \\to 90^\\circ\\)), and correctly identifies the physical roles of the forces exerted by the face on the block?"
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url: "https://nerd-notes.com/ubq/124142/"
date_modified: "2026-09-28T13:30:43+00:00"
---

# A cart on a horizontal track accelerates to the right with a constant acceleration of magnitude \(a\). A small block of mass \(m\) is held against the front face of the cart, which is inclined at an angle \(\theta\) above the horizontal (\(0 < \theta \le 90^\circ\)). The coefficient of static friction between the block and the inclined face is \(\mu_s\), such that the minimum acceleration required to prevent the block from slipping down the face is given by:
\[
a_{\text{min}} = g\left(\dfrac{\sin\theta – \mu_s \cos\theta}{\cos\theta + \mu_s \sin\theta}\right)
\]
Which of the following correctly describes the limiting behavior of \(a_{\text{min}}\) as the face becomes vertical (\(\theta \to 90^\circ\)), and correctly identifies the physical roles of the forces exerted by the face on the block?

A cart on a horizontal track accelerates to the right with a constant acceleration of magnitude \(a\). A small block of mass \(m\) is held against the front face of the cart, which is inclined at an angle \(\theta\) above the horizontal (\(0 < \theta \le 90^\circ\)). The coefficient of static friction between the block and the inclined face is \(\mu_s\), such that the minimum acceleration required to prevent the block from slipping down the face is given by:
\[
a_{\text{min}} = g\left(\dfrac{\sin\theta - \mu_s \cos\theta}{\cos\theta + \mu_s \sin\theta}\right)
\]
Which of the following correctly describes the limiting behavior of \(a_{\text{min}}\) as the face becomes vertical (\(\theta \to 90^\circ\)), and correctly identifies the physical roles of the forces exerted by the face on the block?

![A two-dimensional side-view schematic showing a cart on a horizontal track. A horizontal line at the bottom represents the track. The cart is represented by a rectangular body with two circular wheels resting on the horizontal track. The rightmost face of the cart consists of a planar surface tilted upward and to the left at an angle \(\theta\) relative to the horizontal track, with the angle labeled \(\theta\) between the horizontal track and the inclined surface. A small solid rectangular block of mass \(m\) is in contact with the outer surface of this inclined face. A single horizontal arrow labeled \(a\) is positioned above the cart, pointing toward the right. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790602243-SwiVyH.jpg)

- **A.** \(a_{\text{min}} \to \mu_s g\) ; the normal force acts vertically to balance the gravitational force on the block, while static friction provides the horizontal force required to accelerate the block.
- **B.** \(a_{\text{min}} \to \dfrac{g}{\mu_s}\) ; the normal force acts purely horizontally to provide the block's acceleration, while static friction acts vertically to balance the gravitational force.
- **C.** \(a_{\text{min}} \to \dfrac{g}{\mu_s}\) ; the normal force acts vertically to balance the gravitational force on the block, while static friction acts horizontally to provide the block's acceleration.
- **D.** \(a_{\text{min}} \to \infty\) ; because a vertical surface exerts a purely horizontal normal force, the normal force cannot generate any static friction to prevent slipping.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124142/*