---
title: "Forces and the Principle of Inertia"
book: "High School Physics"
subject: physics
language: en
chapter: 6
exercises: 15
source: https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia
---

# Chapter 6 — Forces and the Principle of Inertia

Stop pedaling and the bicycle coasts to a halt; stop pulling and the sled stands still. Experience whispers that motion must be fed by an effort — and for two thousand years physics agreed. This chapter builds the [force](#def-g10-inertia-force), an arrow measured in [newtons](#def-g10-inertia-force), and states the first great law of mechanics: left truly alone, a moving body keeps its motion, straight and steady, forever.

## 6.1 Modeling an action as a force

**Definition 6.1 (Force).**

A *force* models a mechanical action (a push, a pull, an attraction) exerted on a body. It is characterized by:

- its *point of application* ;
- its direction — the line along which it acts and the way along that line;
- its magnitude, measured in *newtons* (symbol $\mathrm{N}$ ).

A force is drawn as an arrow $\vect F$ (a vector, as in the mathematics volume) from its point of application, with length proportional to the magnitude at a stated scale.

**Example 6.2 (The weight of a schoolbag).**

A schoolbag of mass $m = 4.0\,\mathrm{kg}$ is pulled downward by the Earth with its [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) ([Chapter 4](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#ch-g10-universal-gravitation)): $P = mg = 4.0 \times 9.81 \approx 39\,\mathrm{N}$ — one [newton](#def-g10-inertia-force) is about an apple’s [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight). At the scale $1\,\mathrm{cm} \leftrightarrow
10\,\mathrm{N}$: a vertical arrow $3.9\,\mathrm{cm}$ long, applied at the bag’s center of gravity, pointing down.

## 6.2 An inventory of forces

**Definition 6.3 (Contact and distance forces).**

A *contact force* is exerted at the points where two bodies touch. A *force at a distance* needs no contact: [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) — the gravitational pull of [Chapter 4](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#ch-g10-universal-gravitation) — is the only one used in this chapter.

**Definition 6.4 (The common contact forces).**

Four [contact forces](#def-g10-inertia-contact) cover most everyday situations:

- the *normal reaction* $\vect R$ of a support, perpendicular to its surface, pushing away from it;
- the *tension* $\vect T$ of a wire, rope or chain, directed along it, pulling toward it;
- *friction* $\vect f$ from a surface, the air or water, opposing the sliding or the motion;
- the *thrust* of an engine, jet or hand, pushing along the effort.

![The book on the table: weight P and normal reaction R — same line, equal lengths, opposite ways.](https://one-course.com/images/onecourse/chapters/physics-2/g10-inertia/fig-c8a17ee1dbd0.svg)

*The book on the table: [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) $\vect P$ and [normal reaction](#def-g10-inertia-inventory) $\vect R$ — same line, equal lengths, opposite ways.*

**Example 6.5 (Inventory on a towed sled).**

A sled towed across the snow: [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) $\vect P$ (distance), [normal reaction](#def-g10-inertia-inventory) $\vect R$ of the snow, [tension](#def-g10-inertia-inventory) $\vect T$ of the rope, [friction](#def-g10-inertia-inventory) $\vect f$ opposing the sliding. Nothing else touches the sled: the inventory is complete — *[weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) always, then one [contact force](#def-g10-inertia-contact) per touching object*.

## 6.3 The principle of inertia

**Definition 6.6 (Straight uniform motion).**

A body is in *straight uniform motion* in a frame when its [trajectory](https://one-course.com/books/physics/2/en/chapter/5-relative-motion#def-g10-relative-motion-trajectory) is a straight line traveled at constant speed: its velocity vector $\vect v$ is the same at every instant. Rest is the special case $\vect v = \vect 0$.

**Definition 6.7 (Compensating forces).**

[Forces](#def-g10-inertia-force) acting on a body *compensate* when their combined effect is nil: tip to tail, their arrows return to the start. Two [forces](#def-g10-inertia-force) compensate exactly when they share the same line of action with equal magnitudes and opposite ways.

**Theorem 6.8 (Principle of inertia).**

In the [ground frame](https://one-course.com/books/physics/2/en/chapter/5-relative-motion#def-g10-relative-motion-frames), a body subject to no [force](#def-g10-inertia-force), or to [forces](#def-g10-inertia-force) that [compensate](#def-g10-inertia-compensate), stays at rest or keeps a [straight uniform motion](#def-g10-inertia-uniform); conversely, a body at rest or in [straight uniform motion](#def-g10-inertia-uniform) is subject to [compensating forces](#def-g10-inertia-compensate) (or to none).

**Proof.** *Admitted at this level.* ∎

**Remark 6.9.**

No experiment can fully isolate a body, so the principle is a postulate — glimpsed by Galileo, stated by Newton. A later chapter upgrades it to a law relating [forces](#def-g10-inertia-force) to *changes* of velocity; the Year 1 volume asks in which frames it actually holds.

**Proposition 6.10 (Reading motion from forces, and back).**

In the [ground frame](https://one-course.com/books/physics/2/en/chapter/5-relative-motion#def-g10-relative-motion-frames):

1. if the velocity of a body changes — in magnitude *or* in direction — the [forces](#def-g10-inertia-force) on it do *not* [compensate](#def-g10-inertia-compensate) ;
2. if a body is at rest or in [straight uniform motion](#def-g10-inertia-uniform) , its inventoried [forces](#def-g10-inertia-force) must cancel — determining unknowns.

**Proof.** The first statement is the contrapositive of [Theorem 6.8](#thm-g10-inertia-principle), the second its converse part applied to a completed inventory. ∎

![A hockey puck after release: positions at equal time intervals, one identical velocity arrow each — straight uniform motion.](https://one-course.com/images/onecourse/chapters/physics-2/g10-inertia/fig-0f3f2d2a2222.svg)

*A hockey puck after release: positions at equal time intervals, one identical velocity arrow each — [straight uniform motion](#def-g10-inertia-uniform).*

**Example 6.11 (The hockey puck).**

Released by the stick, a puck crosses $30\,\mathrm{m}$ of smooth ice in $2.5\,\mathrm{s}$, straight at the constant $v = 30/2.5 = 12\,\mathrm{m}/\mathrm{s}$. Inventory: [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) and [normal reaction](#def-g10-inertia-inventory) ([friction](#def-g10-inertia-inventory) negligible). They [compensate](#def-g10-inertia-compensate), and the principle predicts exactly the observed motion: nothing pushes the puck forward — and nothing needs to.

**Remark 6.12 (Galileo against Aristotle).**

Aristotle taught that every motion needs a mover, and daily life — [friction](#def-g10-inertia-inventory) in every contact — seems to agree. Galileo idealized the [friction](#def-g10-inertia-inventory) away: ever smoother surfaces need ever less push to *maintain* motion. Persistence, not rest, is the natural state.

**Remark 6.13 (The lurching passenger).**

When a bus brakes sharply, standing passengers pitch forward. No [force](#def-g10-inertia-force) throws them: by inertia their bodies keep their velocity while the bus loses its own. The seat belt supplies the backward [force](#def-g10-inertia-force) the inventory lacks.

![A curling stone at equal time intervals. Left: P and R compensate — equal steps. Right: an uncompensated friction f remains — the steps shrink, the velocity changes.](https://one-course.com/images/onecourse/chapters/physics-2/g10-inertia/fig-16baabf3365a.svg)

![A curling stone at equal time intervals. Left: P and R compensate — equal steps. Right: an uncompensated friction f remains — the steps shrink, the velocity changes.](https://one-course.com/images/onecourse/chapters/physics-2/g10-inertia/fig-4c610ed68fd5.svg)

*A curling stone at equal time intervals. Left: $\vect P$ and $\vect R$ [compensate](#def-g10-inertia-compensate) — equal steps. Right: an uncompensated [friction](#def-g10-inertia-inventory) $\vect f$ remains — the steps shrink, the velocity changes.*

## 6.4 Balanced forces: statics off a drawing

**Definition 6.14 (Force diagram).**

The *force diagram* of a body shows the body alone, with every [force](#def-g10-inertia-force) acting *on* it drawn to scale from its [point of application](#def-g10-inertia-force); [forces](#def-g10-inertia-force) exerted *by* the body never appear.

**Method 6.15 (Analysing a static situation).**

1. Choose the body to study; work in the [ground frame](https://one-course.com/books/physics/2/en/chapter/5-relative-motion#def-g10-relative-motion-frames) .
2. Inventory: [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) always, one [contact force](#def-g10-inertia-contact) per contact.
3. Draw the [force diagram](#def-g10-inertia-diagram) .
4. The body is at rest, so the [forces](#def-g10-inertia-force) [compensate](#def-g10-inertia-compensate) ( [Theorem 6.8](#thm-g10-inertia-principle) ): read unknown magnitudes off the drawing (equal lengths in 1D, components in 2D).

**Example 6.16 (The book on the table).**

A book of mass $0.50\,\mathrm{kg}$ rests on a table: [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) $P = 0.50 \times 9.81 \approx 4.9\,\mathrm{N}$ and [normal reaction](#def-g10-inertia-inventory) $\vect R$. At rest, $\vect R$ [compensates](#def-g10-inertia-compensate) $\vect P$: vertical, upward, $R = 4.9\,\mathrm{N}$. Add a second book and $R$ adjusts — a support pushes exactly as hard as needed.

![The hanging lamp: the two tensions’ horizontal parts cancel by symmetry; their vertical parts together carry the weight.](https://one-course.com/images/onecourse/chapters/physics-2/g10-inertia/fig-bc9a21917d2c.svg)

*The hanging lamp: the two [tensions](#def-g10-inertia-inventory)’ horizontal parts cancel by symmetry; their vertical parts together carry the [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight).*

**Example 6.17 (The lamp on two wires).**

A lamp of mass $3.0\,\mathrm{kg}$ ($P \approx 29.4\,\mathrm{N}$) hangs from two wires, each at $45^\circ$ from the vertical. By symmetry the [tensions](#def-g10-inertia-inventory) are equal ($T_1 = T_2 = T$) and their horizontal components cancel; the vertical components, each $T \cos 45^\circ$, carry the [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight):

$$
2\,T \cos 45^\circ = P
\qquad\text{so}\qquad
T = \frac{29.4}{2 \times 0.707} \approx 21\,\mathrm{N}
$$

— more than half the [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) per wire: pulling at an angle costs [tension](#def-g10-inertia-inventory).

## 6.5 Exercises

**Exercise 6.1 ★.**

Inventory and classify (contact or distance) the [forces](#def-g10-inertia-force) on: (a) a book at rest on a table; (b) a lamp hanging from its wire; (c) a puck gliding on smooth ice; (d) an apple falling (air resistance negligible).

**Solution of Exercise 6.1.**

(a) [Weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) (distance), [normal reaction](#def-g10-inertia-inventory) of the table (contact). (b) [Weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) (distance), [tension](#def-g10-inertia-inventory) of the wire (contact). (c) [Weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) (distance), [normal reaction](#def-g10-inertia-inventory) of the ice (contact); [friction](#def-g10-inertia-inventory) negligible. (d) [Weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) only — nothing touches the apple.

**Exercise 6.2 ★.**

Compute the [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) of a $250\,\mathrm{g}$ book and of a $60\,\mathrm{kg}$ person. At the scale $1\,\mathrm{cm} \leftrightarrow 1\,\mathrm{N}$, how long is the book’s arrow, and why is the scale hopeless for the person?

**Solution of Exercise 6.2.**

Book: $P = 0.250 \times 9.81 \approx 2.45\,\mathrm{N}$, an arrow about $2.5\,\mathrm{cm}$ long. Person: $P = 60 \times 9.81 \approx 589\,\mathrm{N}$ — an arrow of $5.89\,\mathrm{m}$. Pick a scale to fit the largest [force](#def-g10-inertia-force) on the page.

**Exercise 6.3 ★.**

On a diagram at the scale $1\,\mathrm{cm} \leftrightarrow 2\,\mathrm{N}$, a [force](#def-g10-inertia-force) is a horizontal arrow $3.5\,\mathrm{cm}$ long pointing left, applied at a point $A$. Give its three characteristics.

**Solution of Exercise 6.3.**

[Point of application](#def-g10-inertia-force) $A$; direction horizontal, pointing left; magnitude $3.5 \times 2 = 7.0\,\mathrm{N}$.

**Exercise 6.4 ★.**

A dictionary of mass $1.2\,\mathrm{kg}$ lies on a shelf. Draw its [force diagram](#def-g10-inertia-diagram) and give the magnitude and direction of each [force](#def-g10-inertia-force).

**Solution of Exercise 6.4.**

[Weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) $\vect P$: vertical, downward, $P = 1.2 \times 9.81 \approx
11.8\,\mathrm{N}$. [Normal reaction](#def-g10-inertia-inventory) $\vect R$ of the shelf: same line, upward, $R = 11.8\,\mathrm{N}$ (the dictionary is at rest, so the two [compensate](#def-g10-inertia-compensate)).

**Exercise 6.5 ★.**

True or false — correct the false ones: (a) a body always ends up stopping unless a [force](#def-g10-inertia-force) keeps pushing it; (b) [straight uniform motion](#def-g10-inertia-uniform) requires no uncompensated [force](#def-g10-inertia-force); (c) a body at rest is subject to no [force](#def-g10-inertia-force); (d) if a body’s velocity changes, the [forces](#def-g10-inertia-force) on it do not [compensate](#def-g10-inertia-compensate).

**Solution of Exercise 6.5.**

(a) False: [friction](#def-g10-inertia-inventory) stops things; with no [force](#def-g10-inertia-force) at all the body keeps its [straight uniform motion](#def-g10-inertia-uniform). (b) True (principle of inertia). (c) False: it is subject to [forces](#def-g10-inertia-force) that *[compensate](#def-g10-inertia-compensate)* (e.g. [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) and reaction). (d) True — the contrapositive reading of the principle.

**Exercise 6.6 ★★.**

After release, a puck covers $24\,\mathrm{m}$ of ice in $3.0\,\mathrm{s}$, straight at constant speed. Compute its speed; do the [forces](#def-g10-inertia-force) on it [compensate](#def-g10-inertia-compensate)? On concrete the same puck stops within [metres](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit): what changed in the inventory, and what does [Proposition 6.10](#prop-g10-inertia-converse) conclude?

**Solution of Exercise 6.6.**

*1.* $v = 24/3.0 = 8.0\,\mathrm{m}/\mathrm{s}$. [Forces](#def-g10-inertia-force): [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) and [normal reaction](#def-g10-inertia-inventory); they [compensate](#def-g10-inertia-compensate) ([straight uniform motion](#def-g10-inertia-uniform), [friction](#def-g10-inertia-inventory) negligible). *2.* [Friction](#def-g10-inertia-inventory) from the concrete is no longer negligible: the inventory gains an uncompensated backward [force](#def-g10-inertia-force), and indeed the velocity changes — the puck slows and stops.

**Exercise 6.7 ★★.**

A $70\,\mathrm{kg}$ passenger stands in an elevator rising at the constant $1.5\,\mathrm{m}/\mathrm{s}$. List the [forces](#def-g10-inertia-force) on the passenger; do they [compensate](#def-g10-inertia-compensate)? Give the magnitude of the floor’s [force](#def-g10-inertia-force). Would anything change with the elevator at rest?

**Solution of Exercise 6.7.**

[Weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) $P = 70 \times 9.81 \approx 687\,\mathrm{N}$ down; [force](#def-g10-inertia-force) from the floor up. [Straight uniform motion](#def-g10-inertia-uniform), so they [compensate](#def-g10-inertia-compensate): the floor pushes with $687\,\mathrm{N}$. Nothing changes at rest — rest and uniform motion give the same [force diagram](#def-g10-inertia-diagram).

**Exercise 6.8 ★★.**

A lamp of mass $1.8\,\mathrm{kg}$ hangs at rest from a single vertical wire. Compute the [tension](#def-g10-inertia-inventory) of the wire. The wire is replaced by two *vertical* wires sharing the load equally: what is the [tension](#def-g10-inertia-inventory) of each?

**Solution of Exercise 6.8.**

*1.* At rest, [tension](#def-g10-inertia-inventory) [compensates](#def-g10-inertia-compensate) [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight): $T = 1.8 \times 9.81 \approx 17.7\,\mathrm{N}$. *2.* Each vertical wire carries half: $T = 8.8\,\mathrm{N}$.

**Exercise 6.9 ★★.**

A bus brakes sharply and a standing passenger lurches forward; in a curve at constant speed, the same passenger leans outward. Explain both without inventing a “forward” or “outward” [force](#def-g10-inertia-force).

**Solution of Exercise 6.9.**

Braking: the passenger’s body keeps its velocity (inertia) while the bus loses its own, so the passenger moves forward *relative to the bus*. Turning: the body tends to continue straight while the bus turns underneath, so it drifts toward the outside of the curve. In both cases the “[force](#def-g10-inertia-force)” is fictitious; what is missing is a real [force](#def-g10-inertia-force) (grip, handrail) to change the passenger’s velocity along with the bus’s.

**Exercise 6.10 ★★.**

A parachutist of total mass $85\,\mathrm{kg}$ descends [vertically](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) at the constant $5.0\,\mathrm{m}/\mathrm{s}$. Draw the [force diagram](#def-g10-inertia-diagram) and compute the air resistance on the canopy.

**Solution of Exercise 6.10.**

[Straight uniform motion](#def-g10-inertia-uniform), so [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) and air resistance [compensate](#def-g10-inertia-compensate): drag $= P = 85 \times 9.81 \approx 834\,\mathrm{N}$, vertical and upward.

**Exercise 6.11 ★★.**

A water-skier is towed straight at constant speed; the horizontal rope pulls with $300\,\mathrm{N}$. What is the water’s horizontal drag on the skis (justify)? Which two vertical [forces](#def-g10-inertia-force) [compensate](#def-g10-inertia-compensate) each other?

**Solution of Exercise 6.11.**

*1.* [Straight uniform motion](#def-g10-inertia-uniform): horizontal [forces](#def-g10-inertia-force) [compensate](#def-g10-inertia-compensate), so the drag is $300\,\mathrm{N}$, opposite the rope. *2.* The skier’s [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) and the upward push of the water on the skis.

**Exercise 6.12 ★★★.**

A lamp of mass $2.4\,\mathrm{kg}$ hangs from two wires at $40^\circ$ from the vertical ($\cos 40^\circ \approx 0.766$). Compute each [tension](#def-g10-inertia-inventory). Why does the [tension](#def-g10-inertia-inventory) grow as the wires approach the horizontal, and why could two exactly horizontal wires never hold the lamp?

**Solution of Exercise 6.12.**

*1.* $P = 2.4 \times 9.81 \approx 23.5\,\mathrm{N}$; the vertical components carry the [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight): $2T\cos 40^\circ = P$, so $T = 23.5/(2 \times 0.766) \approx 15.4\,\mathrm{N}$. *2.* $T = P/(2\cos\theta)$ grows without bound as $\theta \to 90^\circ$ since $\cos\theta \to 0$; exactly horizontal wires have *no* vertical component at all, so nothing would carry the [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight).

**Exercise 6.13 ★★★.**

A curling stone is pushed, released, glides while slowing very gradually, then rests. For each phase (push, glide, rest), draw the [force diagram](#def-g10-inertia-diagram) and state whether the [forces](#def-g10-inertia-force) [compensate](#def-g10-inertia-compensate), quoting [Theorem 6.8](#thm-g10-inertia-principle) or [Proposition 6.10](#prop-g10-inertia-converse).

**Solution of Exercise 6.13.**

Push: [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight), reaction, [thrust](#def-g10-inertia-inventory) of the hand, small [friction](#def-g10-inertia-inventory); the speed increases, so the [forces](#def-g10-inertia-force) do not [compensate](#def-g10-inertia-compensate) ([Proposition 6.10](#prop-g10-inertia-converse)). Glide: [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight), reaction, small [friction](#def-g10-inertia-inventory); the stone slows, so again no compensation — [friction](#def-g10-inertia-inventory) is the uncompensated leftover (on ideal ice it would glide forever). Rest: [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) and reaction alone, compensating ([Theorem 6.8](#thm-g10-inertia-principle)).

**Exercise 6.14 ★★★.**

The probe *Voyager 1* coasts through interstellar space at $17\,\mathrm{km}/\mathrm{s}$, engines off. Why does it need no fuel to keep its speed, and what would Aristotle predict? How far does it travel in a year, in $\mathrm{km}$ then in [astronomical units](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-lightyear) ($1\,\mathrm{au} \approx
1.496 \times 10^{8}\,\mathrm{km}$)? Why must it fire a thruster even to change direction *without* changing speed?

**Solution of Exercise 6.14.**

*1.* By the principle of inertia, keeping a constant velocity requires no [force](#def-g10-inertia-force) at all — and in interstellar space almost none acts. Aristotle would predict the probe stops when the “mover” quits. *2.* One year $\approx 3.156 \times 10^{7}\,\mathrm{s}$, so $d = 17 \times 3.156 \times 10^{7} \approx 5.4 \times 10^{8}\,\mathrm{km} \approx
3.6\,\mathrm{au}$. *3.* Velocity is a vector: changing its direction changes the velocity, which requires an uncompensated [force](#def-g10-inertia-force).

**Exercise 6.15 ★★★.**

Three ropes pull on a ring at rest. On a grid where one square is $1\,\mathrm{N}$, two [forces](#def-g10-inertia-force) read $\vect F_1\,(3, 2)$ and $\vect F_2\,(-1, 2)$. Find the third [force](#def-g10-inertia-force)’s components, magnitude and direction.

**Solution of Exercise 6.15.**

At rest the three [forces](#def-g10-inertia-force) [compensate](#def-g10-inertia-compensate): $\vect F_3 = -(\vect F_1 + \vect F_2)$, so $\vect F_3\,(-2, -4)$, magnitude $\sqrt{4 + 16} = \sqrt{20} \approx 4.5\,\mathrm{N}$, pointing opposite the resultant of the first two (down-left on the grid).

## 6.6 Problem: Inertia in three acts

**Problem 6.1.**

Weekend problem — the airport walkway, the ice rink and the parachute: constant velocity as the signature of compensated forces, and why a parachute does not reduce the drag but the speed

Three scenes — a suitcase on a moving walkway, a puck on an ice rink, a parachutist in the evening air — and one law talking in all three.

**Part I — The airport walkway.** A $23\,\mathrm{kg}$ suitcase stands on a walkway moving at the constant speed $0.75\,\mathrm{m}/\mathrm{s}$.

1. Ground-frame motion of the suitcase? Inventory the [forces](#def-g10-inertia-force) on it.
2. Do these [forces](#def-g10-inertia-force) [compensate](#def-g10-inertia-compensate) ? Justify; compute both magnitudes.
3. Compare its [force diagram](#def-g10-inertia-diagram) with the same suitcase’s on the floor. What deep statement does the comparison illustrate?
4. The walkway jerks to a halt. Describe what the suitcase does, without inventing a forward [force](#def-g10-inertia-force) .
5. Its owner walks at $1.2\,\mathrm{m}/\mathrm{s}$ relative to the walkway. Her ground-frame speed? Do the [forces](#def-g10-inertia-force) on *her* [compensate](#def-g10-inertia-compensate) ?

**Part II — The ice rink.**

6. A $160\,\mathrm{g}$ puck glides straight at the constant $8.0\,\mathrm{m}/\mathrm{s}$ . Draw its [force diagram](#def-g10-inertia-diagram) with magnitudes.
7. Aristotle claims the puck needs a mover. What does the rink reply, and why do everyday surfaces seem to side with him?
8. The puck crosses a rough patch and slows. Conclusion about the [forces](#def-g10-inertia-force) there? Which [force](#def-g10-inertia-force) is the culprit?
9. A skater curves smoothly at constant speed. Do the [forces](#def-g10-inertia-force) on her [compensate](#def-g10-inertia-compensate) ? Careful: what kind of quantity is velocity?
10. Match diagram to motion: (a) compensated [forces](#def-g10-inertia-force) ; (b) an extra [force](#def-g10-inertia-force) opposite $\vect v$ ; (c) an extra [force](#def-g10-inertia-force) perpendicular to $\vect v$ — (i) slowing straight; (ii) turning at constant speed; (iii) [straight uniform motion](#def-g10-inertia-uniform) .

**Part III — The parachute.** A jumper’s total mass, gear included, is $80\,\mathrm{kg}$.

11. Leaving the plane, vertical speed and drag are still zero. Compute the [forces](#def-g10-inertia-force) ; do they [compensate](#def-g10-inertia-compensate) ? What must happen next?
12. Drag grows with speed. Describe the fall while drag $<$ [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) .
13. Before the canopy opens, the jumper reaches a steady $50\,\mathrm{m}/\mathrm{s}$ (“terminal speed”). Compute the drag. What is the motion, and why is it “inertia in disguise”?
14. The canopy opens: the drag now far exceeds the [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) . What happens to the velocity? (The jumper moves *down* throughout.)
15. A new steady $5.0\,\mathrm{m}/\mathrm{s}$ is reached. Compute the drag; compare with question 13.
16. The punchline: what did the parachute change — the terminal drag, or the speed at which the drag reaches the [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) ?

**Part IV — Verdicts.**

17. Complete and justify: “at rest or in [straight uniform motion](#def-g10-inertia-uniform) $\iff$ …”. Why does rest deserve no separate law?
18. In which act would Aristotle’s “motion needs a mover” have failed most visibly, and what would Galileo point at?
19. An exam script claims: “the parachutist falls at constant speed, so no [force](#def-g10-inertia-force) acts on him”. Correct it in one line.
20. Finale: for each act, name the pair of [forces](#def-g10-inertia-force) that [compensate](#def-g10-inertia-compensate) , and state the one law used twenty times in this problem.

**Solution of Problem 6.1.**

**1.** [Straight uniform motion](#def-g10-inertia-uniform) at $0.75\,\mathrm{m}/\mathrm{s}$. [Forces](#def-g10-inertia-force): [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) and the [normal reaction](#def-g10-inertia-inventory) of the belt (no [friction](#def-g10-inertia-inventory) needed: the suitcase does not slide on the belt). **2.** Yes — [straight uniform motion](#def-g10-inertia-uniform) ([Theorem 6.8](#thm-g10-inertia-principle)). $P = 23 \times 9.81 \approx
226\,\mathrm{N}$, so $R = 226\,\mathrm{N}$. **3.** The diagrams are identical. Rest and [straight uniform motion](#def-g10-inertia-uniform) are one and the same mechanical state: inertia does not distinguish them. **4.** The suitcase keeps its $0.75\,\mathrm{m}/\mathrm{s}$ while the belt stops: it slides or tips forward until [friction](#def-g10-inertia-inventory) absorbs its motion. Nothing pushed it — it merely kept the velocity it had. **5.** Speeds along the same direction add: $0.75 + 1.2 =
1.95\,\mathrm{m}/\mathrm{s}$. Her velocity is constant, so the [forces](#def-g10-inertia-force) on her [compensate](#def-g10-inertia-compensate). **6.** $P = 0.160 \times 9.81 \approx 1.57\,\mathrm{N}$ down, $R = 1.57\,\mathrm{N}$ up, and *no* horizontal [force](#def-g10-inertia-force). **7.** The puck keeps $8.0\,\mathrm{m}/\mathrm{s}$ with no mover whatsoever. Everyday surfaces hide [friction](#def-g10-inertia-inventory) — an uncompensated backward [force](#def-g10-inertia-force) — so motion seems to die by itself. **8.** Its velocity changes, so the [forces](#def-g10-inertia-force) no longer [compensate](#def-g10-inertia-compensate) ([Proposition 6.10](#prop-g10-inertia-converse)); the culprit is the [friction](#def-g10-inertia-inventory) of the rough ice. **9.** No: velocity is a *vector*, and its direction is changing, so the [forces](#def-g10-inertia-force) do not [compensate](#def-g10-inertia-compensate) — the ice pushes sideways on her blades. **10.** (a)–(iii), (b)–(i), (c)–(ii). **11.** [Weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) $P = 80 \times 9.81 \approx 785\,\mathrm{N}$, drag zero: nothing [compensates](#def-g10-inertia-compensate) the [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight), so the velocity must change — the jumper speeds up downward. **12.** While drag $<$ [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) the [forces](#def-g10-inertia-force) still do not [compensate](#def-g10-inertia-compensate): the speed keeps growing, and the drag grows with it. **13.** Steady speed: [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) and drag [compensate](#def-g10-inertia-compensate), so the drag is $785\,\mathrm{N}$. The motion is straight and uniform — a falling body in the exact mechanical state of the suitcase on the walkway: inertia in disguise. **14.** Drag now exceeds [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight): the [forces](#def-g10-inertia-force) do not [compensate](#def-g10-inertia-compensate), so the velocity changes — the jumper, still descending, slows down; as the speed drops, the drag shrinks back toward the [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight). **15.** Steady again: drag $= P = 785\,\mathrm{N}$ — exactly the same as at $50\,\mathrm{m}/\mathrm{s}$. **16.** The terminal drag is always equal to the [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight), canopy or not. The parachute changes the *speed* at which the drag reaches $785\,\mathrm{N}$: $5\,\mathrm{m}/\mathrm{s}$ instead of $50\,\mathrm{m}/\mathrm{s}$. **17.** “…the [forces](#def-g10-inertia-force) acting on the body [compensate](#def-g10-inertia-compensate) (or none act)” — the two readings of [Theorem 6.8](#thm-g10-inertia-principle). Rest is just $\vect v = \vect 0$, a [straight uniform motion](#def-g10-inertia-uniform) like any other. **18.** The rink: the puck cruises with no mover in sight. Galileo points at the ever-smoother surface — [friction](#def-g10-inertia-inventory), not nature, is what stops things. **19.** [Forces](#def-g10-inertia-force) do act — [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) $785\,\mathrm{N}$ down and drag $785\,\mathrm{N}$ up; they *[compensate](#def-g10-inertia-compensate)*, which is precisely why the speed is constant. **20.** Walkway: [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) and [normal reaction](#def-g10-inertia-inventory). Rink: [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) and [normal reaction](#def-g10-inertia-inventory). Parachute: [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) and drag. The one law: the principle of inertia, in its direct and converse readings.