---
title: "Work of a Force"
book: "High School Physics"
subject: physics
language: en
chapter: 17
exercises: 15
source: https://one-course.com/books/physics/2/en/chapter/17-work-of-a-force
---

# Chapter 17 — Work of a Force

Push a wardrobe across the room and you have worked; push against a wall all afternoon and, whatever your muscles report, physics counts zero. [Work](#def-g11-work-of-force-work) is what a [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) delivers *along a motion* — [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) in transit ([Chapter 9](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#ch-g10-energy-conservation)), paid in [joules](#def-g11-work-of-force-work). This chapter prices the [forces](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) of [Chapter 16](https://one-course.com/books/physics/2/en/chapter/16-forces-and-motion#ch-g11-forces-and-motion) — pulls at an angle, [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight), [friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory) — and clocks the payment rate: [power](#def-g11-work-of-force-power).

## 17.1 Work of a constant force

**Definition 17.1 (Work of a constant force).**

A constant [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) $\vect F$ applied to an object moving along a straight segment from $A$ to $B$ does the *work*

$$
W_{AB}(\vect F) = F \times AB \times \cos\theta ,
$$

where $F$ is the magnitude of the [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) (in $\mathrm{N}$), $AB$ the length of the displacement (in $\mathrm{m}$) and $\theta$ the angle between $\vect F$ and $\vect{AB}$. Work is measured in *joules* ($\mathrm{J}$): one joule is the work of a [one-newton](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) along one [metre](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit) of its own direction.

**Remark 17.2.**

$F \times AB \times \cos\theta$ is exactly the scalar product $\vect F \cdot \vect{AB}$ of this year’s mathematics volume. Only the component $F\cos\theta$ *along* the motion earns anything.

![Only the projection F of the force on the displacement works: W_AB( F) = F × AB ×.](https://one-course.com/images/onecourse/chapters/physics-2/g11-work-of-force/fig-cbe772a7790b.svg)

*Only the projection $F\cos\theta$ of the [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) on the displacement [works](#def-g11-work-of-force-work): $W_{AB}(\vect F) = F \times AB \times \cos\theta$.*

**Example 17.3 (Suitcase on a leash).**

A traveller drags a suitcase $300\,\mathrm{m}$ across a terminal, the strap pulled with $F = 40\,\mathrm{N}$ at $\theta = 50{}^{\circ}$ to the floor: $W = 40 \times 300 \times \cos50{}^{\circ} \approx
7.7\,\mathrm{kJ}$. Pulled flat ($\theta = 0$) the same [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) would deliver $12\,\mathrm{kJ}$: a third of the pull lifts nothing.

**Definition 17.4 (Motor, resistive, zero work).**

The sign of $\cos\theta$ sorts every [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) into three regimes:

- $\theta < 90{}^{\circ}$ : $W > 0$ , the [work](#def-g11-work-of-force-work) is *motor* — the [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) feeds [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) into the motion;
- $\theta = 90{}^{\circ}$ : $W = 0$ , the [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) [works](#def-g11-work-of-force-work) not at all, however large;
- $\theta > 90{}^{\circ}$ : $W < 0$ , the [work](#def-g11-work-of-force-work) is *resistive* — the [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) drains [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) . The pure case is a brake at $180{}^{\circ}$ : $800\,\mathrm{N}$ over $45\,\mathrm{m}$ takes $800 \times 45 \times (-1)  = -36\,\mathrm{kJ}$ out of a car, into hot discs.

![The three regimes — motor, zero, resistive — by the angle between force and displacement (red arrow).](https://one-course.com/images/onecourse/chapters/physics-2/g11-work-of-force/fig-f85719ec6d36.svg)

![The three regimes — motor, zero, resistive — by the angle between force and displacement (red arrow).](https://one-course.com/images/onecourse/chapters/physics-2/g11-work-of-force/fig-ffa1e2cd548a.svg)

![The three regimes — motor, zero, resistive — by the angle between force and displacement (red arrow).](https://one-course.com/images/onecourse/chapters/physics-2/g11-work-of-force/fig-25729cb1b7cc.svg)

*The three regimes — [motor](#def-g11-work-of-force-sign), zero, [resistive](#def-g11-work-of-force-sign) — by the angle between [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) and displacement (red arrow).*

## 17.2 The work of the weight

**Proposition 17.5 (Work of the weight).**

When an object of mass $m$ moves from $A$ (altitude $z_A$) to $B$ (altitude $z_B$) along *any* path made of straight segments, its [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) $\vect P$ does the [work](#def-g11-work-of-force-work)

$$
W_{AB}(\vect P) = mg\,(z_A - z_B) = \pm\, mgh ,
$$

where $h = \abs{z_A - z_B}$ is the height drop: $+mgh$ going down, $-mgh$ going up — the drop alone matters, not the route.

**Proof.** On one straight segment $AB$, the [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) (magnitude $mg$, straight down) makes an angle $\theta$ with $\vect{AB}$, and $AB\cos\theta$ is precisely the projection of the displacement on the downward vertical: $z_A - z_B$. Hence $W = mg \times AB\cos\theta = mg\,(z_A - z_B)$. Along a chain of segments the [works](#def-g11-work-of-force-work) add and the drops telescope: $mg(z_A - z_C) + mg(z_C - z_B) = mg(z_A - z_B)$. ∎

**Proposition 17.6 (Any path).**

The same formula holds along any curved path from $A$ to $B$.

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

**Remark 17.7.**

Plausible — a curve is as close as we like to a broken line of many short segments — but making “as close as we like” honest is calculus, done in [Chapter 29](https://one-course.com/books/physics/2/en/chapter/29-work-and-mechanical-energy#ch-g12-work-and-energy).

![Straight (blue) or wandering (green), every path from A to B drops the same h: the weight does +mgh on both.](https://one-course.com/images/onecourse/chapters/physics-2/g11-work-of-force/fig-ff9981f7e2e4.svg)

*Straight (blue) or wandering (green), every path from $A$ to $B$ drops the same $h$: the [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) does $+mgh$ on both.*

**Example 17.8 (Hiker).**

A $78\,\mathrm{kg}$ hiker (pack included) climbs $450\,\mathrm{m}$, by whichever trail: $W(\vect P) = -78 \times 9.81 \times 450 \approx -344\,\mathrm{kJ}$, and exactly $+344\,\mathrm{kJ}$ back down, whatever the detours.

## 17.3 The work of friction

**Proposition 17.9 (Work of sliding friction).**

A [friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory) [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) of constant magnitude $f$, at every instant opposite the motion, does the [work](#def-g11-work-of-force-work) $W = -fL$ along a path of total length $L$: always [resistive](#def-g11-work-of-force-sign), and proportional to the *length of the path travelled* — not to the displacement.

**Proof.** Each short straight piece has the [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) at $180{}^{\circ}$ to the motion, contributing $-f \times (\text{its length})$; the pieces add up to $-fL$. A curved path is chopped into short pieces likewise, the limit admitted as for the [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) ([Remark 17.7](#rem-g11-work-of-force-anypath)). ∎

**Remark 17.10 (Friction against weight: the great divide).**

The [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight)’s [work](#def-g11-work-of-force-work) forgets the path; [friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory)’s is a toll per [metre](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit). On a round trip the [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) nets zero — what it takes uphill it refunds downhill — while [friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory) charges $-fL$ on every leg and refunds nothing: the [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight)’s [work](#def-g11-work-of-force-work) can be stored and recovered (potential [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy), [Chapter 18](https://one-course.com/books/physics/2/en/chapter/18-mechanical-energy-and-its-conservation#ch-g11-mechanical-energy)), [friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory)’s is lost as heat.

**Example 17.11 (There and back).**

A crate slides $5.0\,\mathrm{m}$ across a floor and $5.0\,\mathrm{m}$ back, against $f = 60\,\mathrm{N}$: [friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory) does $-60 \times 10 =
-600\,\mathrm{J}$ even though the crate ends where it started. [Weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) and normal [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force), perpendicular to the motion: zero throughout.

**Remark 17.12 (What about a spring?).**

A spring pulls harder the further it is stretched: its [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) is not constant and [Definition 17.1](#def-g11-work-of-force-work) does not apply. Its [work](#def-g11-work-of-force-work) means cutting the stretch into infinitesimal steps and summing — calculus, delivered in [Chapter 29](https://one-course.com/books/physics/2/en/chapter/29-work-and-mechanical-energy#ch-g12-work-and-energy).

## 17.4 Power

**Definition 17.13 (Power).**

The *power* of a [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) is the rate at which it [works](#def-g11-work-of-force-work), $P = W/\Delta t$, in *watts* ($\mathrm{W} = \mathrm{J}/\mathrm{s}$) — the same watt as in [Chapter 12](https://one-course.com/books/physics/2/en/chapter/12-electric-circuits-and-power#ch-g11-circuits-and-power). A pre-SI [unit](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit) survives on engine badges: the *horsepower*, $1\,\mathrm{hp} = 736\,\mathrm{W}$.

**Proposition 17.14 (Power at constant velocity).**

If the object moves at constant speed $v$, a constant [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) $\vect F$ at angle $\theta$ to the motion delivers the [power](#def-g11-work-of-force-power) $P = F v \cos\theta$ — in particular $P = Fv$ for a [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) along the motion.

**Proof.** In a time $\Delta t$ the object covers $d = v\,\Delta t$, so $W = F \times v\,\Delta t \times \cos\theta$; divide by $\Delta t$. ∎

**Example 17.15 (Crane).**

A crane hoists a $600\,\mathrm{kg}$ pallet at a steady $0.90\,\mathrm{m}/\mathrm{s}$: the cable’s [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) equals the [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight), $mg = 5.89 \times 10^{3}\,\mathrm{N}$, so $P = Fv = 5890 \times 0.90 \approx 5.3\,\mathrm{kW} \approx 7.2\,\mathrm{hp}$ — a seven-horse team in a steel box.

**Method 17.16 (Climbing with less force: the zigzag).**

To raise a load a height $h$ with a limited [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force):

1. The bill is fixed: by [Proposition 17.5](#prop-g11-work-of-force-weight) , any route to the top costs the [work](#def-g11-work-of-force-work) $mgh$ .
2. Stretch the path: on a slope of length $L$ the [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) needed (constant speed, [friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory) aside) is $F = mgh/L$ — double the length, halve the [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) .
3. Pay the surcharge: [friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory) charges by the [metre](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit) ( [Proposition 17.9](#prop-g11-work-of-force-friction) ), adding $fL$ .

Mountain roads zigzag for step 2: same height, less [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force), same [work](#def-g11-work-of-force-work).

![Hauling 1200\, kg up h = 300\, m: the product F × L = mgh = 3.53\, MJ is fixed, so the road trades length for force (see ).](https://one-course.com/images/onecourse/chapters/physics-2/g11-work-of-force/fig-0b0e8124801e.svg)

*Hauling $1200\,\mathrm{kg}$ up $h = 300\,\mathrm{m}$: the product $F \times L = mgh = 3.53\,\mathrm{MJ}$ is fixed, so the road trades length for [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) (see [Exercise 17.12](#exo-g11-work-of-force-12)).*

## 17.5 Exercises

**Exercise 17.1 ★.**

A constant [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) $F = 40\,\mathrm{N}$ acts over a straight $50\,\mathrm{m}$ displacement: compute its [work](#def-g11-work-of-force-work) for $\theta = 0{}^{\circ}$, $60{}^{\circ}$, $90{}^{\circ}$ and $120{}^{\circ}$.

**Solution of Exercise 17.1.**

$W = 40 \times 50 \times \cos\theta$: $+2000\,\mathrm{J}$; $+1000\,\mathrm{J}$; $0$; $-1000\,\mathrm{J}$.

**Exercise 17.2 ★.**

A $12\,\mathrm{kg}$ toolbox is lowered $2.5\,\mathrm{m}$ from a van, then later raised back. [Work](#def-g11-work-of-force-work) of its [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) in each case?

**Solution of Exercise 17.2.**

$mgh = 12 \times 9.81 \times 2.5 = 294\,\mathrm{J}$: lowered, $+294\,\mathrm{J}$ ([weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) along the drop); raised, $-294\,\mathrm{J}$.

**Exercise 17.3 ★.**

[Motor](#def-g11-work-of-force-sign), [resistive](#def-g11-work-of-force-sign) or zero? (a) the [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) of a falling apple; (b) the [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) of a rising ball; (c) the normal [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) on a sliding box; (d) [friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory) on a braking bicycle; (e) the string’s [tension](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory) on a swinging pendulum bob.

**Solution of Exercise 17.3.**

(a) [motor](#def-g11-work-of-force-sign); (b) [resistive](#def-g11-work-of-force-sign); (c) zero (perpendicular); (d) [resistive](#def-g11-work-of-force-sign); (e) zero — the string’s [tension](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory) stays perpendicular to the bob’s motion.

**Exercise 17.4 ★.**

A crane lifts $600\,\mathrm{kg}$ by $25\,\mathrm{m}$ at constant speed. [Work](#def-g11-work-of-force-work) of the lifting [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) and of the [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight)? What is their sum, and why?

**Solution of Exercise 17.4.**

Constant speed: lifting [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) $= mg = 5.89 \times 10^{3}\,\mathrm{N}$, so $W_{\text{lift}} = 5886 \times 25 \approx +147\,\mathrm{kJ}$ and $W(\vect P) = -147\,\mathrm{kJ}$. Sum zero: at constant velocity the [forces](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) balance ([Chapter 16](https://one-course.com/books/physics/2/en/chapter/16-forces-and-motion#ch-g11-forces-and-motion)), so their [works](#def-g11-work-of-force-work) cancel.

**Exercise 17.5 ★.**

An escalator [motor](#def-g11-work-of-force-sign) does $90\,\mathrm{kJ}$ of [work](#def-g11-work-of-force-work) every minute. Its [power](#def-g11-work-of-force-power) in [watts](#def-g11-work-of-force-power)? In [horsepower](#def-g11-work-of-force-power)?

**Solution of Exercise 17.5.**

$P = 9.0 \times 10^{4}/60 = 1.5\,\mathrm{kW}$; $1500/736 \approx 2.0\,\mathrm{hp}$.

**Exercise 17.6 ★★.**

A sled is towed $200\,\mathrm{m}$ by a rope of [tension](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory) $120\,\mathrm{N}$ at $30{}^{\circ}$ above the snow; [friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory) is $45\,\mathrm{N}$. [Work](#def-g11-work-of-force-work) of each of the four [forces](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) (rope, [friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory), [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight), normal) and total?

**Solution of Exercise 17.6.**

Rope: $120 \times 200 \times \cos30{}^{\circ} \approx
+20.8\,\mathrm{kJ}$; [friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory): $-45 \times 200 = -9.0\,\mathrm{kJ}$; [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) and normal [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force), perpendicular: $0$. Total $+11.8\,\mathrm{kJ}$: the sled gains speed.

**Exercise 17.7 ★★.**

A cyclist rides at a constant $25\,\mathrm{km}/\mathrm{h}$ against a total drag of $18\,\mathrm{N}$. [Power](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-power) delivered? [Work](#def-g11-work-of-force-work) done in one hour?

**Solution of Exercise 17.7.**

$v = 25/3.6 = 6.94\,\mathrm{m}/\mathrm{s}$; $P = Fv = 18 \times 6.94 \approx
125\,\mathrm{W}$. In one hour, $W = P\,\Delta t = 125 \times 3600 =
450\,\mathrm{kJ}$ (equivalently $18 \times 25\,000\,\mathrm{m}$).

**Exercise 17.8 ★★.**

A $65\,\mathrm{kg}$ skier descends $120\,\mathrm{m}$ of height, by a $800\,\mathrm{m}$ steep slope or a $1500\,\mathrm{m}$ gentle track; [friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory) on the skis is $30\,\mathrm{N}$ on both. [Work](#def-g11-work-of-force-work) of the [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) and of [friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory) on each route? Which [work](#def-g11-work-of-force-work) cares about the route?

**Solution of Exercise 17.8.**

[Weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight), both routes: $+65 \times 9.81 \times 120 \approx
+76.5\,\mathrm{kJ}$ (drop only). [Friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory): $-30 \times 800 =
-24\,\mathrm{kJ}$ on the slope, $-30 \times 1500 = -45\,\mathrm{kJ}$ on the track. Only [friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory) cares about the route.

**Exercise 17.9 ★★.**

A $20\,\mathrm{kg}$ box is pushed $4.0\,\mathrm{m}$ up a $25{}^{\circ}$ ramp at constant speed, against $30\,\mathrm{N}$ of [friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory). Compute the [works](#def-g11-work-of-force-work) of the [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) and of [friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory), then (from the constant speed) the [work](#def-g11-work-of-force-work) and magnitude of the push, parallel to the ramp.

**Solution of Exercise 17.9.**

Rise $h = 4.0\sin25{}^{\circ} = 1.69\,\mathrm{m}$: $W(\vect P) = -20 \times 9.81 \times 1.69 \approx -332\,\mathrm{J}$; $W_f = -30 \times 4.0 = -120\,\mathrm{J}$. Constant speed: total [work](#def-g11-work-of-force-work) zero, so $W_{\text{push}} = +452\,\mathrm{J}$ and $F = 452/4.0 \approx 113\,\mathrm{N}$.

**Exercise 17.10 ★★.**

A car cruises at $130\,\mathrm{km}/\mathrm{h}$ with its engine delivering $50\,\mathrm{kW}$ to the wheels. What total [resistive](#def-g11-work-of-force-sign) [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) does it fight?

**Solution of Exercise 17.10.**

$v = 130/3.6 = 36.1\,\mathrm{m}/\mathrm{s}$; constant speed, so $F = P/v = 5.0 \times 10^{4}/36.1 \approx 1.4\,\mathrm{kN}$.

**Exercise 17.11 ★★.**

A canal horse tows a barge at $1.0\,\mathrm{m}/\mathrm{s}$, the rope of [tension](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory) $800\,\mathrm{N}$ making $20{}^{\circ}$ with the towpath. Compute the horse’s [power](#def-g11-work-of-force-power), in [watts](#def-g11-work-of-force-power) and in [horsepower](#def-g11-work-of-force-power). Comment.

**Solution of Exercise 17.11.**

$P = Fv\cos\theta = 800 \times 1.0 \times \cos20{}^{\circ}
\approx 752\,\mathrm{W}$; in [horsepower](#def-g11-work-of-force-power), $752/736 \approx 1.02$ — almost exactly what a strong horse sustains, the comparison Watt built his [unit](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit) on.

**Exercise 17.12 ★★★.**

A $1200\,\mathrm{kg}$ car climbs $h = 300\,\mathrm{m}$ to a pass, by a straight track of $20\%$ grade or a zigzag road of $6\%$ grade (grade $=
\sin\alpha$). Neglecting [friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory), at constant speed, compute for each route the length, the [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) needed along the slope, and its [work](#def-g11-work-of-force-work). Conclude in one sentence.

**Solution of Exercise 17.12.**

Lengths $L = h/\text{grade}$: $1500\,\mathrm{m}$ and $5000\,\mathrm{m}$. [Forces](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) $F = mg \times \text{grade}$: $1200 \times 9.81 \times 0.20 \approx
2.35\,\mathrm{kN}$ and $0.71\,\mathrm{kN}$. [Works](#def-g11-work-of-force-work) $FL$: $2354 \times 1500
\approx 3.53\,\mathrm{MJ}$ and $706 \times 5000 \approx 3.53\,\mathrm{MJ}$ — both equal $mgh$: the road trades [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) for length, never [work](#def-g11-work-of-force-work).

**Exercise 17.13 ★★★.**

A winch raises $250\,\mathrm{kg}$ of tiles $15\,\mathrm{m}$ up scaffolding at a steady $0.40\,\mathrm{m}/\mathrm{s}$. Compute the [work](#def-g11-work-of-force-work) of the cable’s [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force), the duration, the mechanical [power](#def-g11-work-of-force-power) (twice: $W/\Delta t$ and $Fv$), and the [electrical power](https://one-course.com/books/physics/2/en/chapter/12-electric-circuits-and-power#prop-g11-circuits-and-power-power) drawn at $70\%$ [motor](#def-g11-work-of-force-sign) [efficiency](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-efficiency).

**Solution of Exercise 17.13.**

$F = mg = 2.45 \times 10^{3}\,\mathrm{N}$; $W = 2453 \times 15 \approx 36.8\,\mathrm{kJ}$; $\Delta t = 15/0.40 = 37.5\,\mathrm{s}$; $P = 36800/37.5 \approx 981\,\mathrm{W} = 2453 \times 0.40$ ($Fv$ agrees); electrical: $981/0.70 \approx 1.4\,\mathrm{kW}$.

**Exercise 17.14 ★★★.**

A wardrobe is slid against $f = 140\,\mathrm{N}$ of [friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory) from one corner of a room to the opposite one: straight along the $6.0\,\mathrm{m}$ diagonal, or $3.6\,\mathrm{m}$ then $4.8\,\mathrm{m}$ along the walls. [Friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory) [work](#def-g11-work-of-force-work) on each route, then on a diagonal round trip? What does the [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) do on that round trip, and what deep difference between the two [forces](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) does this expose?

**Solution of Exercise 17.14.**

Diagonal: $-140 \times 6.0 = -840\,\mathrm{J}$; walls: $-140 \times 8.4 \approx -1.18\,\mathrm{kJ}$; round trip: $-140 \times 12 = -1.68\,\mathrm{kJ}$. The [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) does zero on the round trip (no net drop). [Friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory)’s [work](#def-g11-work-of-force-work) depends on the path and never refunds; the [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight)’s depends only on the endpoints ([Remark 17.10](#rem-g11-work-of-force-divide)).

**Exercise 17.15 ★★★.**

A cyclist ($80\,\mathrm{kg}$ with bike) climbs a pass: $600\,\mathrm{m}$ of height over $12\,\mathrm{km}$ of road, at a constant $15\,\mathrm{km}/\mathrm{h}$, against $14\,\mathrm{N}$ of drag and rolling [resistance](https://one-course.com/books/physics/2/en/chapter/12-electric-circuits-and-power#def-g11-circuits-and-power-resistance). Compute the [work](#def-g11-work-of-force-work) supplied against gravity, against [friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory), and in total; the duration of the climb; the average [power](#def-g11-work-of-force-power), in [watts](#def-g11-work-of-force-power) and [horsepower](#def-g11-work-of-force-power).

**Solution of Exercise 17.15.**

Gravity: $80 \times 9.81 \times 600 \approx 471\,\mathrm{kJ}$; [friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory): $14 \times 12000 = 168\,\mathrm{kJ}$; total $639\,\mathrm{kJ}$. $v = 4.17\,\mathrm{m}/\mathrm{s}$, so $\Delta t = 12000/4.17 = 2880\,\mathrm{s} =
48\,\mathrm{min}$ and $P = 6.39 \times 10^{5}/2880 \approx 222\,\mathrm{W} \approx
0.30\,\mathrm{hp}$ — a strong amateur, a third of a horse.

## 17.6 Problem: The mover’s day

**Problem 17.1.**

Weekend problem — physics pays by the joule: a day of sliding, hoisting and hairpin bends, ending with the humbling discovery of what a kettle thinks of honest labour

A mover’s crew empties a warehouse, hauls the load up to a third-floor flat, then drives the van to a hilltop village. Every task is priced in [joules](#def-g11-work-of-force-work); the kitchen kettle will audit the day. Boxes: $30\,\mathrm{kg}$.

**Part I — Across the warehouse floor.** A box is pushed with a horizontal [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) $F = 110\,\mathrm{N}$ over $d = 8.0\,\mathrm{m}$; sliding [friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory) is $f = 95\,\mathrm{N}$.

1. Compute the [work](#def-g11-work-of-force-work) of the push.
2. Compute the [work](#def-g11-work-of-force-work) of [friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory) , of the [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) , and of the normal [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) (justify the zeros).
3. Total [work](#def-g11-work-of-force-work) on the box, and its sign’s meaning (the box started at rest)? What push keeps the speed constant ( [Chapter 16](https://one-course.com/books/physics/2/en/chapter/16-forces-and-motion#ch-g11-forces-and-motion) )?
4. The far corner can be reached along the $10.0\,\mathrm{m}$ diagonal or along two walls ( $6.0\,\mathrm{m}$ then $8.0\,\mathrm{m}$ ). [Friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory) [work](#def-g11-work-of-force-work) on each route?
5. On a diagonal round trip (out and back), compute the [work](#def-g11-work-of-force-work) of [friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory) and the [work](#def-g11-work-of-force-work) of the [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) . Which [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) refunds?

**Part II — Up to the third floor ($h = 9.0\,\mathrm{m}$).**

6. Compute the [work](#def-g11-work-of-force-work) of the [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) on a box raised to the flat. What [work](#def-g11-work-of-force-work) must the crew supply at least?
7. Carried up the stairs by a $70\,\mathrm{kg}$ mover, gravity charges for mover *and* box: total [work](#def-g11-work-of-force-work) against gravity, and the fraction actually spent on the box?
8. A trolley ramp is $30\,\mathrm{m}$ long for the same $9.0\,\mathrm{m}$ rise, with $40\,\mathrm{N}$ of rolling [friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory) . At constant speed, compute the push needed along the ramp and its [work](#def-g11-work-of-force-work) .
9. Compare the ramp with a straight vertical hoist: [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) needed, [work](#def-g11-work-of-force-work) done. What does the ramp buy, and at what price?
10. A rope-and-pulley hoist raises the box at $0.50\,\mathrm{m}/\mathrm{s}$ . Duration and mechanical [power](#def-g11-work-of-force-power) ?
11. Recover that [power](#def-g11-work-of-force-power) as $P = Fv$ . The hoist’s [motor](#def-g11-work-of-force-sign) is $60\%$ efficient: [electrical power](https://one-course.com/books/physics/2/en/chapter/12-electric-circuits-and-power#prop-g11-circuits-and-power-power) drawn?

**Part III — The zigzag road.** The loaded van ($3500\,\mathrm{kg}$) must reach a village $240\,\mathrm{m}$ above the valley: straight lane, $1.0\,\mathrm{km}$ at $24\%$ grade, or paved road, $8.0\,\mathrm{km}$ at $3.0\%$ grade (grade $= \sin\alpha$).

12. Check that both routes climb the same $240\,\mathrm{m}$ .
13. At constant speed and neglecting [friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory) , compute the driving [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) needed along each route.
14. [Work](#def-g11-work-of-force-work) of the driving [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) on each route? Compare with $mgh$ .
15. Rolling [resistance](https://one-course.com/books/physics/2/en/chapter/12-electric-circuits-and-power#def-g11-circuits-and-power-resistance) is $400\,\mathrm{N}$ on both: compute the extra [work](#def-g11-work-of-force-work) on each route. What does the zigzag’s comfort cost?
16. At $36\,\mathrm{km}/\mathrm{h}$ , compute the engine [power](#def-g11-work-of-force-power) needed on each route (in $\mathrm{kW}$ and $\mathrm{hp}$ ). Why do roads zigzag?

**Part IV — The day’s ledger.**

17. The crew slides $60$ boxes across the floor (the push of question 1) and hoists all $60$ up the $9.0\,\mathrm{m}$ (question 6). Total mechanical [work](#def-g11-work-of-force-work) delivered to the boxes?
18. Add the stair carrier’s own body: $20$ trips up at $70\,\mathrm{kg}$ (downhill refunds go to hot knees, not to the ledger). Day’s total?
19. The crew worked $8.0$ hours: average mechanical [power](#def-g11-work-of-force-power) ? How long would the $2.0\,\mathrm{kW}$ kettle take to expend the same [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) ?
20. Finale: in two sentences, say what the [joule](#def-g11-work-of-force-work) pays for — and does not (the wall pushed all afternoon, the box held motionless) — and why movers own winches, ramps and zigzag roads rather than bigger muscles.

**Solution of Problem 17.1.**

**1.** $W = 110 \times 8.0 = +880\,\mathrm{J}$.

**2.** [Friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory): $-95 \times 8.0 = -760\,\mathrm{J}$. [Weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) and normal [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) are perpendicular to the motion: zero each.

**3.** Total $+120\,\mathrm{J}$: net [motor work](#def-g11-work-of-force-sign), the box speeds up. A $95\,\mathrm{N}$ push balances [friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory) — zero [net force](https://one-course.com/books/physics/2/en/chapter/16-forces-and-motion#def-g11-forces-and-motion-netforce), constant speed, zero total [work](#def-g11-work-of-force-work).

**4.** Diagonal: $-95 \times 10.0 = -950\,\mathrm{J}$; walls: $-95 \times 14.0 = -1.33\,\mathrm{kJ}$: [friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory) charges by the [metre](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit).

**5.** [Friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory): $-95 \times 20.0 = -1.9\,\mathrm{kJ}$; [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight): $0$ (no net drop). Only the [weight](https://one-course.com/books/physics/2/en/chapter/4-universal-gravitation-and-weight#def-g10-universal-gravitation-weight) refunds.

**6.** $W(\vect P) = -30 \times 9.81 \times 9.0 =
-2.65\,\mathrm{kJ}$; the crew must supply at least $+2.65\,\mathrm{kJ}$.

**7.** $(70 + 30) \times 9.81 \times 9.0 = 8.83\,\mathrm{kJ}$; the box’s share is $2.65/8.83 \approx 30\%$ — carrying is mostly self-transport.

**8.** $\sin\alpha = 9.0/30 = 0.30$: $F = mg\sin\alpha + f = 88.3 + 40 = 128\,\mathrm{N}$; $W = 128 \times 30 = 3.85\,\mathrm{kJ}$ ($= 2.65 + 1.2$).

**9.** Vertical hoist: $F = mg = 294\,\mathrm{N}$, $W = 2.65\,\mathrm{kJ}$. The ramp buys a smaller [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) ($128$ against $294\,\mathrm{N}$) at the price of more [work](#def-g11-work-of-force-work): [friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory)’s $1.2\,\mathrm{kJ}$ toll.

**10.** $\Delta t = 9.0/0.50 = 18\,\mathrm{s}$; $P = 2649/18 \approx 147\,\mathrm{W}$.

**11.** $P = Fv = 294 \times 0.50 = 147\,\mathrm{W}$ — same. Electrical: $147/0.60 \approx 245\,\mathrm{W}$.

**12.** $1000 \times 0.24 = 240\,\mathrm{m}$ and $8000 \times 0.030 = 240\,\mathrm{m}$: same summit.

**13.** $F = mg \times \text{grade}$: lane $3500 \times 9.81 \times 0.24 \approx 8.2\,\mathrm{kN}$; road $\approx 1.0\,\mathrm{kN}$.

**14.** $8240 \times 1000 \approx 8.24\,\mathrm{MJ}$ and $1030 \times 8000 \approx 8.24\,\mathrm{MJ}$ — both $= mgh = 3500
\times 9.81 \times 240$.

**15.** Rolling [resistance](https://one-course.com/books/physics/2/en/chapter/12-electric-circuits-and-power#def-g11-circuits-and-power-resistance): $400 \times 1000 = 0.40\,\mathrm{MJ}$ against $400 \times 8000 = 3.2\,\mathrm{MJ}$: the gentle road’s comfort costs $2.8\,\mathrm{MJ}$ of extra [friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory).

**16.** $v = 10\,\mathrm{m}/\mathrm{s}$. Road: $(1030 + 400) \times 10 = 14.3\,\mathrm{kW} \approx 19\,\mathrm{hp}$; lane: $(8240 + 400) \times 10 = 86.4\,\mathrm{kW} \approx 117\,\mathrm{hp}$ — beyond a loaded van. Roads zigzag so that modest engines (and tyres) can climb: same [work](#def-g11-work-of-force-work), spread [thin](https://one-course.com/books/physics/2/en/chapter/10-lenses-images-and-the-eye#def-g11-lenses-and-eye-lens).

**17.** Per box: $880 + 2649 = 3.53\,\mathrm{kJ}$; for $60$ boxes, $60 \times 3529 \approx 212\,\mathrm{kJ}$.

**18.** Stairs: $20 \times 70 \times 9.81 \times 9.0 \approx
124\,\mathrm{kJ}$. Day’s total $\approx 336\,\mathrm{kJ} = 0.34\,\mathrm{MJ}$.

**19.** $P = 3.36 \times 10^{5}/28800 \approx 12\,\mathrm{W}$. The kettle: $3.36 \times 10^{5}/2000 = 168\,\mathrm{s}$ — under three minutes for the whole day’s labour.

**20.** The [joule](#def-g11-work-of-force-work) pays for [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) delivered *along* a motion — nothing for the wall pushed or the box held, however exhausting (muscles burn [chemical energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-forms) even doing zero mechanical [work](#def-g11-work-of-force-work)). Since human [power](#def-g11-work-of-force-power) output is a few dozen [watts](#def-g11-work-of-force-power), movers let machines reshape the [force](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-force) — winch, ramp, zigzag — while the bill $mgh$ never changes.