Torque — Physica

Equivalent forms

\vec{\tau} = \vec{r} \times \vec{F}

\tau = I\alpha

The rotational analogue of force — and the reason wrenches have long handles.

Symbol	Name	SI unit	Dimension	Range
$τ$	Torqueoutput Twisting effect of a force about an axis	N·m	ML^2T^-2	0 – 1000
$r$	Lever Arm Distance from pivot to point where force is applied	m	L	0 – 5
$F$	Force Magnitude of applied force	N	MLT^-2	0 – 500
$θ$	Angle Angle between lever arm and force vector	rad	1	0 – 3.14159

Unit systems

Symbol	SI	CGS	Imperial
$τ$	N·m	dyn·cm	ft·lbf
$r$	m	cm	ft
$F$	N	dyn	lbf
$θ$	rad	rad	rad

Where it holds

Valid for any rigid body about any chosen axis. Net torque equals

I\alpha

(Newton's second law for rotation) for rigid rotation.

Dimensional analysis

\tau \to [ML^{2}T^{-2}]

r\cdot F \to [L]\cdot [MLT^{-2}] = [ML^{2}T^{-2}]

Both sides are

N\cdot m

(dimensionally Joules, but called

N\cdot m

for torque).

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Discovery

Archimedes · -250

Archimedes formalized the law of the lever: 'Give me a place to stand and I will move the Earth'. Later integrated by Newton and Euler into rotational dynamics.

Try this

Why is it harder to open a door by pushing near the hinge than near the handle?

Compare the torque from pushing 20 N on a 0.8 m wide door at the handle vs. 0.1 m from the hinge. Apply τ = rF sin θ with θ = 90°.

Research status: stable

Real-world applications

Torque wrenches for tightening bolts
Engine torque ratings
Bicycle gear ratios
Wind turbine and helicopter rotor design

Common misconceptions

Torque is not energy despite sharing units $(N\cdot m)$ — never call it Joules
The lever arm is the perpendicular distance from pivot to the force line
Increasing r isn't free — you trade displacement for force

Experimental verification

Torque wrenches in mechanical engineering measure

\tau to \pm 2

% accuracy. Lab demos with meter sticks and spring scales confirm

rF \sin \theta

.

Derivation

From the work-energy theorem applied to a rotating body:

dW = F\cdot dx = F\cdot (r d\theta )\cdot \sin \theta

.

Defining

\tau \equiv dW/d\theta

gives

\tau = rF \sin \theta

.

Vector form:

\tau = r \times F

.

Limiting cases

r \to 0

⟶

\tau \to 0

A force at the pivot produces no rotation.

\theta \to 0

⟶

\tau \to 0

Force parallel to lever arm produces no torque (just pulls or pushes the pivot).

\theta = \pi /2

⟶

\tau \to rF (\max )

Perpendicular force gives maximum torque for given r and F.

What if…

What if you use a longer wrench?

Torque scales linearly with r. Doubling wrench length halves the force needed for the same torque.

What if you push at

30^{\circ}

instead of

90^{\circ}

?

$\tau$ scales by $\sin \theta$ . $\sin (30^{\circ}) = 0.5$ , so you lose half the effective torque.

What if multiple torques act on the body?

Net torque $= sum$ of all torques (with sign). Equilibrium requires $\Sigma \tau = 0$ about every axis.

1

Door at hinge vs handle

Given ·

F:: 20
r handle:: 0.8
r hinge:: 0.1
θ:: 1.5708

Find ·

\tau

Steps

Handle: $\tau = 0.8 \times 20 \times \sin (90^{\circ}) = 16 N\cdot m$
Near hinge: $\tau = 0.1 \times 20 \times 1 = 2 N\cdot m$
Same force, $8\times$ less rotational effect — that's why handles are placed far from hinges

Result ·

\tau

_handle

= 16 N\cdot m

,

\tau

_hinge

= 2 N\cdot m

—

8\times

harder near hinge

2

Loosening a stuck bolt

Given ·

F:: 200
r:: 0.3
θ:: 1.5708

Find ·

\tau

Steps

Use a 30 cm wrench: $r = 0.3\,\mathrm{m}$
Push perpendicular with 200 N
$\tau = 0.3 \times 200 \times 1 = 60 N\cdot m$ — typical car lug-nut spec

Result ·

\tau = 60 N\cdot m

Newton's Second Law→Rotational Kinetic Energy→Angular Momentum→