---
title: "An hourglass of total mass \\(M\\) rests on a sensitive force scale that measures the upward normal force \\(F_N\\) exerted on the hourglass. At time \\(t = 0\\), sand begins to fall from rest from the upper bulb at a constant mass flow rate; the first grains strike the base of the lower bulb at \\(t_1\\), the last grains exit the upper bulb at \\(t_2\\), and all sand comes to rest at \\(t_3\\). The graph shows the scale reading \\(F_N\\) as a function of time \\(t\\). Which of the following statements correctly interprets the relationship between the graph and the physical behavior of the hourglass system?"
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url: "https://nerd-notes.com/ubq/124249/"
date_modified: "2026-09-28T14:04:31+00:00"
---

# An hourglass of total mass \(M\) rests on a sensitive force scale that measures the upward normal force \(F_N\) exerted on the hourglass. At time \(t = 0\), sand begins to fall from rest from the upper bulb at a constant mass flow rate; the first grains strike the base of the lower bulb at \(t_1\), the last grains exit the upper bulb at \(t_2\), and all sand comes to rest at \(t_3\). The graph shows the scale reading \(F_N\) as a function of time \(t\). Which of the following statements correctly interprets the relationship between the graph and the physical behavior of the hourglass system?

An hourglass of total mass \(M\) rests on a sensitive force scale that measures the upward normal force \(F_N\) exerted on the hourglass. At time \(t = 0\), sand begins to fall from rest from the upper bulb at a constant mass flow rate; the first grains strike the base of the lower bulb at \(t_1\), the last grains exit the upper bulb at \(t_2\), and all sand comes to rest at \(t_3\). The graph shows the scale reading \(F_N\) as a function of time \(t\). Which of the following statements correctly interprets the relationship between the graph and the physical behavior of the hourglass system?

![A line graph with a horizontal axis labeled t and a vertical axis labeled F_N. A horizontal dashed line spans the width of the graph at a vertical level labeled Mg. Along the horizontal axis, four time values are marked: 0, t_1, t_2, and t_3, ordered from left to right. A solid curve represents the scale reading. For t < 0, the solid curve lies on the dashed line at Mg. Between t = 0 and t_1, the solid curve slopes downward below the dashed line, reaching a minimum just before t_1, then rises steeply to rejoin the dashed line at t_1. From t_1 to t_2, the solid curve remains flat along the dashed line at Mg. Between t_2 and t_3, the solid curve rises smoothly above the dashed line to form a rounded peak before returning to the dashed line at t_3. For t > t_3, the curve continues horizontally along the dashed line. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604270-LJUEdY.jpg)

- **A.** Between \(t = 0\) and \(t = t_1\), the normal force is less than \(Mg\) because the center of mass of the hourglass system has an upward acceleration as mass exits the upper chamber.
- **B.** The magnitude of the area bounded between the curve and the line \(F_N = Mg\) from \(t = 0\) to \(t_1\) equals the magnitude of the area bounded from \(t_2\) to \(t_3\) because the total change in linear momentum of the system over the entire process is zero.
- **C.** Between \(t = t_1\) and \(t = t_2\), the normal force equals \(Mg\) because the total downward linear momentum of the sand in the hourglass is zero during steady flow.
- **D.** Between \(t = t_2\) and \(t = t_3\), the normal force exceeds \(Mg\) because gravitational attraction between the fallen sand and airborne sand exerts an additional downward external force on the scale.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124249/*