---
title: "A student is investigating the acceleration of a block of mass \\(m\\) sliding on a rough horizontal surface with a coefficient of kinetic friction \\(\\mu_k\\). The student applies a constant force of magnitude \\(F_A\\) to the block at an angle \\(\\theta\\) with the horizontal. Two different methods are used:  – **Case 1:** The block is pulled by a string at an angle \\(\\theta\\) above the horizontal. – **Case 2:** The block is pushed by a rigid rod at an angle \\(\\theta\\) below the horizontal.  In both cases, the block accelerates to the right and remains in contact with the surface."
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url: "https://nerd-notes.com/ubq/109170/"
date_modified: "2026-03-20T13:05:44+00:00"
---

# A student is investigating the acceleration of a block of mass \(m\) sliding on a rough horizontal surface with a coefficient of kinetic friction \(\mu_k\). The student applies a constant force of magnitude \(F_A\) to the block at an angle \(\theta\) with the horizontal. Two different methods are used:

– **Case 1:** The block is pulled by a string at an angle \(\theta\) above the horizontal.
– **Case 2:** The block is pushed by a rigid rod at an angle \(\theta\) below the horizontal.

In both cases, the block accelerates to the right and remains in contact with the surface.

A student is investigating the acceleration of a block of mass \(m\) sliding on a rough horizontal surface with a coefficient of kinetic friction \(\mu_k\). The student applies a constant force of magnitude \(F_A\) to the block at an angle \(\theta\) with the horizontal. Two different methods are used:

- **Case 1:** The block is pulled by a string at an angle \(\theta\) above the horizontal.
- **Case 2:** The block is pushed by a rigid rod at an angle \(\theta\) below the horizontal.

In both cases, the block accelerates to the right and remains in contact with the surface.

![Two side-by-side diagrams. On the left, labeled 'Case 1', a rectangular block rests on a horizontal surface. A straight line representing a string is attached to the top right corner of the block, angling upward and to the right at an angle theta relative to a horizontal dashed line. On the right, labeled 'Case 2', an identical block is on a horizontal surface. A straight line representing a rigid rod pushes against the top left corner of the block, angling downward and to the right at an angle theta relative to a horizontal dashed line.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1774000955-ng6SUM.jpg)

**Part a)** For both Case 1 and Case 2, **draw** and **label** the forces (not components) that are exerted on the block. Each force must be represented by a distinct arrow starting on, and pointing away from, the dot. *(3 points)*

**Part b)** **Derive** an expression for the normal force \(F_{N1}\) exerted on the block in Case 1 and the normal force \(F_{N2}\) exerted on the block in Case 2. Express your answers in terms of \(m\), \(F_A\), \(\theta\), and fundamental constants, as appropriate. *(2 points)*

**Part c)** The block is observed to have a greater acceleration in Case 1 than in Case 2. **Justify** this observation using physical principles and the expressions derived in part (b). *(2 points)*

**Part d)** In a new trial for Case 2, the student adjusts the magnitude of the applied force \(F_A\) so that the block moves to the right with a constant velocity. **Derive** an expression for this required force \(F_A\). Express your answer in terms of \(m\), \(\theta\), \(\mu_k\), and fundamental constants, as appropriate. *(3 points)*

**Part e)** Based on the expression derived in part (d), **predict** what happens to the required force \(F_A\) as the angle \(\theta\) increases such that \(\tan\theta\) approaches \(\dfrac{1}{\mu_k}\). **Explain** the physical reason for this behavior in terms of the vertical and horizontal force components. *(2 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/109170/*