Orbiting planets and pendulum illustrating classical mechanics principles

Torque and rotational dynamics

PHYS 201 · Rotational Mechanics

Torque is the rotational effect of a force. This lesson explains torque, lever arms, rotational equilibrium, and Newton's second law for rotation.

Key equations

\vec{\tau}=\vec{r}\times\vec{F}\tau=rF\sin\theta\tau=F\ell\sum \tau=I\alpha\sum F=ma\sum \tau=0\sum \vec{F}=0\tau=RTRT=I\alphaa=R\alphaW=\int \tau\,d\thetaW=\tau\Delta\thetaP=\tau\omega

Learning objectives

Define torque using cross product and lever arm forms.
Determine torque direction using the right-hand rule.
Apply rotational Newton's second law.
Use torque equilibrium for static systems.

Force versus torque

Force causes linear acceleration. Torque causes angular acceleration. A force's rotational effectiveness depends not only on its magnitude but also on where and in what direction it is applied.

Torque is defined by the cross product

$ec{ au}= ec{r} imes ec{F}$

where $ec{r}$ points from the rotation axis to the point where the force is applied. The magnitude is

$au=rFsin heta$

where $heta$ is the angle between $ec{r}$ and $ec{F}$ .

Lever arm

The lever arm is the perpendicular distance from the rotation axis to the line of action of the force. Torque magnitude can also be written

$au=Fell$

where $ell$ is the lever arm.

This explains why a door is easier to open by pushing far from the hinge and perpendicular to the door. Pushing near the hinge or along the door creates little torque.

Direction of torque

Torque is a vector. Its direction is determined by the right-hand rule. Point your fingers in the direction of $ec{r}$ , curl toward $ec{F}$ , and your thumb points in the direction of $ec{ au}$ .

For rotation in a plane, torque direction is often represented as positive or negative: counterclockwise positive and clockwise negative, or the reverse if chosen consistently.

Rotational Newton's second law

For rotation about a fixed axis, the rotational analog of Newton's second law is

$sum au=Ialpha$

Here $I$ is the moment of inertia and $alpha$ is angular acceleration. This resembles

$sum F=ma$

but rotational inertia depends not only on mass but on how mass is distributed relative to the rotation axis.

Rotational equilibrium

If an object is in rotational equilibrium,

$sum au=0$

If it is also in translational equilibrium,

$sum ec{F}=0$

A ladder leaning against a wall, a balanced seesaw, and a bridge beam can be analyzed using both force and torque equilibrium.

Choosing the pivot

In static equilibrium problems, torque can be calculated about any axis. A smart choice simplifies the equations. Choosing an axis through an unknown force makes that force produce zero torque because $r=0$ . This can eliminate unknowns.

For example, in a seesaw problem, choosing the pivot at the fulcrum eliminates the normal force from the torque equation.

Pulley and rotational dynamics

If a string pulls on a pulley of radius $R$ with tension $T$ , the torque is

$au=RT$

if the tension is tangential. The pulley angular acceleration satisfies

$RT=Ialpha$

If the string does not slip,

$a=Ralpha$

This couples translational motion of the string to rotational motion of the pulley.

Work and power in rotation

Rotational work is

$W=int au,d heta$

For constant torque,

$W= auDelta heta$

Rotational power is

$P= auomega$

These mirror linear relations $W=Fd$ and $P=Fv$ .

The big idea

Torque measures the rotational effect of force. It depends on force, distance from axis, and angle. Rotational dynamics follows $sum au=Ialpha$ , the rotational counterpart of Newton's second law. Torque ideas explain levers, doors, pulleys, seesaws, rolling objects, and rotating machines.

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