The rotational effect of a force

A rigid body is an extended object modelled as keeping the same shape while forces act on it. The phrase “extended object” is doing some work here. A force has a size and direction, yes, but it also acts at a particular place. Push a door near the hinge and it barely turns. Push the same door near the handle and it swings easily.

Torque is the turning effect of a force about a specified rotation axis. For a force acting at a point on a rigid body,

[ au = Fr\sin heta]

where ( au) is the torque about the axis (N m), (F) is the magnitude of the force (N), (r) is the distance from the axis to the point where the force is applied (m), and ( heta) is the angle between the force and the radius line from the axis to the point of application (rad or degrees, treated as dimensionless inside the sine function).

Only the component of the force perpendicular to the radius line produces the turning effect. You can say the same thing another way: torque is force multiplied by the perpendicular distance from the axis to the force’s line of action. In many problems, using that perpendicular distance is the cleanest method.

Sense, not full vector treatment

Torque is formally a vector, but in this course we usually only need its rotational sense: clockwise or counter-clockwise. Choose one sense as positive, then keep that choice throughout the calculation. A sign error in a torque question is usually not a physics error; it’s a bookkeeping error.

Maximum torque occurs when the force is perpendicular to the radius line. If the force points directly through the axis, the torque is zero: a force through the hinge cannot open the door.

Resultant torque zero

A body is in rotational equilibrium if its angular velocity stays constant because the resultant torque on it is zero. It may be stationary, or it may already be rotating at a steady rate. The condition is

[\sum au = 0]

where (\sum) means the algebraic sum of torques, with clockwise and counter-clockwise torques given opposite signs.

Think of it as the rotational version of translational equilibrium. In translation, zero resultant force gives no linear acceleration. In rotation, zero resultant torque gives no angular acceleration.

Principle of moments

The practical rule is the principle of moments: for rotational equilibrium, the total clockwise torque equals the total counter-clockwise torque about the same axis. You can choose the axis wherever it makes the working easier. In statics questions, a neat trick is to put the axis through an unknown force, so that force produces no torque and drops out of the equation.

Centre of mass in rigid-body problems

The centre of mass is the point of an extended body that moves as if the body’s whole mass were concentrated there when considering its linear motion. You don’t need to calculate centres of mass in this topic, but you do need to use the idea correctly.

A force through the centre of mass can produce translational acceleration without rotational acceleration. If the force does not pass through the centre of mass, it generally produces both translation and rotation. Weight is treated as acting at the centre of mass; for a uniform regular object this is often at the geometrical centre, but not always.

Couples

A couple is a pair of equal and opposite forces with different lines of action. Its resultant force is zero, so by itself it does not cause linear acceleration. It does, however, have a non-zero resultant torque, so it tends to cause rotation.

For a couple, the torque is the magnitude of one force multiplied by the perpendicular separation of the two lines of action. Don’t use the distance between the points of application unless that distance is also perpendicular to the forces.

The rotational form of Newton’s first idea

When an extended rigid body has an unbalanced resultant torque acting on it, it has angular acceleration, (\alpha) (rad s(^{-2})). Put more simply: if the clockwise torques and counter-clockwise torques don’t cancel, the rotation changes.

This matches the idea that an unbalanced force causes linear acceleration, but there’s one extra thing to watch in rotation: the distribution of mass. Two objects can have the same mass and still respond very differently to the same torque if one has more mass farther from the axis.

Restoring torque and simple harmonic motion

Torque can act as a restoring effect too. If a twisted spring or torsion wire produces a torque proportional to angular displacement, directed back towards equilibrium, the object can undergo angular simple harmonic motion. It’s the rotational version of a mass on a spring: displacement becomes angular displacement, and force becomes torque.

You’ll meet this again when oscillations are treated formally. For now, keep the link clear: torque changes rotational motion in the same way that force changes translational motion.

Describing rotation

Angular displacement is the angle through which a body rotates about an axis, measured in radians. Angular velocity is how quickly that angular displacement changes. The syllabus uses the term angular velocity, but you do not need a formal vector treatment; in calculations, treat it as angular speed, with a clockwise or counter-clockwise sense when needed.

[\omega = \frac{\Delta heta}{\Delta t}]

where (\omega) is angular speed (rad s(^{-1})), (\Delta heta) is angular displacement (rad), and (\Delta t) is the time interval (s).

Angular acceleration is how quickly angular velocity changes:

[\alpha = \frac{\Delta\omega}{\Delta t}]

where (\Delta\omega) is the change in angular speed (rad s(^{-1})).

Radians are the natural unit

One complete revolution is (2\pi) rad. Before using the equations in this topic, convert revolutions per minute into rad s(^{-1}). Here, unit conversion is not a finishing touch — it is part of the physics.

Linking angular and linear quantities

A point farther from the axis on a rotating body moves faster than a point closer in. The link is

[v=\omega r]

where (v) is the linear speed of a point on the body (m s(^{-1})).

Students often mix up three accelerations:

• angular acceleration, which describes the changing rotation of the whole body;
• tangential acceleration, (a_t), which is the linear acceleration along the tangent: [a_t=\alpha r] where (a_t) is tangential acceleration (m s(^{-2}));
• centripetal acceleration, (a_c), which is the inward acceleration of a point moving in a circle: [a_c=\frac{v^2}{r}=\omega^2r] where (a_c) is centripetal acceleration (m s(^{-2})).

If angular velocity is constant, then (\alpha=0) and (a_t=0). Even so, (a_c) is still present for every point not on the axis.

Where rotation turns up later

The same language describes wheels, satellites, charged particles in magnetic fields and coils rotating in generators. For a satellite or charged particle in circular motion, angular speed tells you how quickly the radius line sweeps round. For a conducting coil, controlled rotation through a magnetic field is the starting point for generating an electric current.

The linear equations, translated carefully

With uniform angular acceleration, the rotational kinematics equations mirror the constant-acceleration equations for straight-line motion. Check units before substituting: angular quantities must be in radians, not revolutions or degrees.

For constant (\alpha):

[\Delta heta = \frac{\omega_f+\omega_i}{2}t]

[\omega_f=\omega_i+\alpha t]

[\Delta heta=\omega_i t+\frac{1}{2}\alpha t^2]

[\omega_f^2=\omega_i^2+2\alpha\Delta heta]

where (\omega_i) is initial angular speed (rad s(^{-1})), (\omega_f) is final angular speed (rad s(^{-1})), and (t) is elapsed time (s).

Linear constant-acceleration quantities and equations matched to their rotational equivalents.

Graphs say the same thing

On an angular velocity–time graph, the gradient gives angular acceleration. The area under the graph gives angular displacement. For a straight-line graph, split the area into a rectangle and a triangle; that gives the equation containing (\frac{1}{2}\alpha t^2). It’s the same idea used for linear motion graphs.

When not to use them

These equations only work for uniform angular acceleration. If the torque changes so that (\alpha) is non-uniform, use a graph, energy, angular momentum, or calculus-style reasoning instead of forcing these equations onto the situation.

The rotational equivalent of mass — with a warning

Moment of inertia tells you how much a body resists a change in its rotational motion about a particular axis. It is a scalar, written (I), with unit kg m(^2).

Mass by itself doesn’t decide the value. Put the same mass close to the axis and the moment of inertia is smaller than if that mass sits farther away. That’s why a skater spins faster when they pull in their arms, and why flywheels often carry much of their mass near the rim.

Axis matters

Moment of inertia is always measured “about an axis”. A rod has one moment of inertia about its centre, and a different one about one end. Change the axis, and (I) usually changes too.

For many uniform shapes, the form is

[I=kMR^2]

where (k) is a dimensionless shape factor, (M) is the total mass of the body (kg), and (R) is a characteristic radius or length (m). You are not expected to memorise formulae for specific continuous mass distributions; they will be provided when needed.

Modelling real bodies

You can model a real wheel as a hoop if most of its mass lies near the rim. If the mass is spread through the face, a solid disc model fits better. The model is not “true” or “false”; it is more or less useful for the required precision.

A simple way to test the idea is to roll objects with the same mass and radius down the same ramp. A hoop, a solid cylinder and a sphere do not accelerate equally, because their moments of inertia differ. With repeated timing, a measured height change of the centre of mass, and uncertainty ranges, you can judge whether your data distinguish one model from another.

Adding point masses

For a system made of point masses, find the moment of inertia by adding the contribution from each mass:

[I=\sum mr^2]

Here, (m) is each point mass in the system (kg). The square matters: if you double the distance from the axis, that mass contributes four times as much to (I).

Take two identical small masses on a light rod, each the same distance from the central axis. Together, they give twice the contribution of one mass. A rod is called light when its own moment of inertia is negligible compared with the attached masses.

Distributed bodies are many point masses

Think of a continuous object as a huge number of tiny point masses. The same idea sits behind the formulae for discs, spheres and rods, although you don’t have to derive those formulae. If an exam gives a specific mass distribution, the appropriate expression for (I) will be supplied when needed.

Torque produces angular acceleration

Newton’s second law for rotation is

[ au=I\alpha]

where ( au) is the average resultant torque if the torque changes over the time interval. It matches the linear idea that a resultant force causes acceleration, except mass is replaced by moment of inertia, and linear acceleration by angular acceleration.

More torque means more angular acceleration. For the same torque, a larger moment of inertia means a smaller angular acceleration. That’s why a light door swings more easily than a heavy fire door, and why rotation becomes harder to change when mass is moved outward.

Translation and rotation together

A force on an extended body can produce linear acceleration of the centre of mass as well as angular acceleration about the centre of mass. When the force’s line of action passes through the centre of mass, there is no torque about the centre of mass. When it misses, there is a torque, so the body starts to rotate.

Action and reaction torques

If body A exerts a torque on body B, body B exerts an equal and opposite torque on body A. Just like ordinary Newton’s third law pairs, these two torques act on different bodies. They don’t cancel when you consider one body on its own.

Rotational momentum

Angular momentum is the rotational momentum of a body. It equals the product of moment of inertia and angular velocity:

[L=I\omega]

where (L) is angular momentum (kg m(^2) s(^{-1})). Since the radian is dimensionless, it is usually left out of the unit.

You do not need the full vector treatment here. Still, you must keep track of clockwise and counter-clockwise sense. If two rotating bodies have equal magnitudes of angular momentum in opposite senses, their total angular momentum can be zero.

A gyroscope gives a good physical picture. A fast-spinning wheel has a large angular momentum, so changing its rotation takes a significant torque. That’s why spinning wheels can feel unexpectedly “stubborn” when you try to twist their axes.

The conservation law

Angular momentum stays constant unless a resultant external torque acts on the system. Internal forces and internal torques may shift angular momentum from one part of the system to another, but the total cannot change.

For a system before and after an interaction,

[\sum I_i\omega_i=\sum I_f\omega_f]

where (I_i) is an initial moment of inertia of a body or component (kg m(^2)) and (I_f) is its final moment of inertia (kg m(^2)).

Changing moment of inertia

If (I) decreases and no external torque acts, (\omega) increases. That’s the ice-skater effect. Pulling the arms in does not create angular momentum; it lowers the moment of inertia, so the angular speed rises to keep angular momentum constant.

For coupled rotating bodies, the signs matter. If two coaxial wheels are clamped together, angular momentum is conserved during the coupling, but rotational kinetic energy may decrease because friction or deformation can transfer mechanical energy into thermal energy. A useful form is

[I_1\omega_1+I_2\omega_2=(I_1+I_2)\Omega]

where (I_1) and (I_2) are the moments of inertia of the two bodies (kg m(^2)), (\omega_1) and (\omega_2) are their initial angular speeds with signs for sense (rad s(^{-1})), and (\Omega) is their common final angular speed (rad s(^{-1})).

Orbits and Bohr’s model

Angular momentum conservation also explains why orbiting bodies speed up when they are closer to the central body and slow down when they are farther away. This gives one route to Kepler’s second law: the radius line sweeps out equal areas in equal times when there is no external torque about the central mass.

In the Bohr model of hydrogen, angular momentum cannot take just any value; it is quantised. When that angular momentum condition is combined with the electrical force providing circular motion, it leads to the allowed orbital radius for the ground state. The model matters historically, even though the modern treatment of the atom is quantum-mechanical.

Charged particles in magnetic fields give another link. The magnetic force can make a particle travel in a circular path, so rotational language helps describe its angular speed and radius.

Torque acting for a time

Angular impulse is the change in angular momentum caused by a resultant torque acting for a time interval:

[\Delta L= au\Delta t=\Delta(I\omega)]

Think of it as the rotational equivalent of impulse in linear mechanics. Its unit may be written as N m s, which is the same as kg m(^2) s(^{-1}), the unit of angular momentum.

Area under a torque–time graph

When torque changes with time, the angular impulse equals the area under the torque–time graph. Positive area increases angular momentum in the chosen positive rotational sense; negative area reduces it, or increases it in the opposite sense.

Changing (I) while rotating

Since (L=I\omega), angular impulse can change angular momentum through angular speed, moment of inertia, or both. With no external torque, (\Delta L=0), so any change in (I) has to be balanced by a compensating change in (\omega). If an external torque acts while (I) is changing as well, treat the two effects together.

That’s why, for extended bodies that change shape, or for systems where two rotating bodies become coupled, it’s usually best to start with angular momentum and deal with kinematics afterwards.