Rotational mechanics is the area of mechanics concerned with objects that are rotating or following curved paths. An example is circular motion. In general the motion of any object can be thought of in terms of a combination of linear and rotational motions.
Rotational kinematics involve describing the motion of rotating objects without understanding the forces that cause them to undergo that motion. It has many similarites to linear motion. Instead of displacement (distance) in linear mechanics, rotational mechanics considers angular displacement which is the angle something has rotated by. Similarly, velocity (rate of change of displacement) becomes angular velocity, the rate of change of angular displacement over time. Normally the Greek letter θ is used to denote angular displacement, an ω (omega) is used to denote angular velocity and α to denote angular acceleration. Angular displacement has units of radians so angular velocity has units of radians per second. These quantities can be related to each other through:
From these definitions, rotational versions of the SUVAT equations can be derived when the angular acceleration is constant. They are:
where θ0 is the initial angular displacement and ω is the final angular velocity.
Vector Nature of Angular Velocity
In order to describe which axis an object is rotating around, angular velocity is a vector quantity. It is defined as having a magnitude equal to the angular speed of the object and points along the axis of rotation defined by the right hand rule. In this way, angular displacement and angular accleration are also vector quantities.
Angular velocity is an example of a pseudovector. This means that although it often behaves like a vector, it does not always behave as a true vector would. An example of this is mirror symmetry. If a vector is reflected in a mirror, it will generally point in a different direction. However, if a rotating objecct is reflected in a mirror, it rotates in the same way and its angular velocity vector points in the same direction as before.
Relation to Linear Mechanics
Suppose an object is rotating about an axis ar some distance r. Suppose it moves a small distance ds, which is equal to r dθ for a small angular displacement dθ. Dividing both sides by dt to produce:
which is simply v=rω. This equation relates the speed of a particle to its angular speed. Furthermore, the acceleration of the particle can be written in terms of a radial component (directed towards the axis of rotation) and a tangential component (perpendicular to the radial component). Differentiating v=rω again produces a relation between the tangential acceleration and angular acceleration: atan=rα. The radial component is the centripetal acceleration and causes the particle to rotate about an axis. It can be expressed as: arad=ω2r.
A key quantity in linear mechancis is momentum. Unsurprisingly, there is a rotational equivalent known as "angular momentum" and this too is a pseudovector. It is defined as the cross product of the position vector of a particle with its linear momentum:
Sometimes it may be written using the letter L. Like linear momentum, it has a conservation law: the sum of the angular momenta of particles in a system will remain constant so long as no external torques act on that system.
Rotational dynamics involves describing the motion of rotating objects by understanding the forces that act upon them. The rotational equivalent of a force is a torque. A torque acting on an object produces an angular acceleration. Torque is comonly denoted as a τ. It is defined as τ=Fl where F is the force and l is the perpendicular distance between the force and the axis of rotation. This can be extended to a vector using the cross product as:
with is the position vector of the object. The moment of inertia is a rotational equivalent to mass in linear mechanics. It is a measure of the resistance to acceleration by a torque about a particular axis and defined as:
A rotational version of Newton's second law can now be expressed as:
More generally, it is expressed in terms of angular momentum:
Energy in Rotating Systems
A rotational equivalent of kinetic energy can be found for rotating objects. It follows a similar form to linear kinetic energy, namely:
where ωf and ωi are the final and initial angular velocities and W is the work done.
Suppose a mass is rotating around some axis at an angular speed ω. Now consider a point particle, with mass dm, that is part of this mass. If it moves at some speed v as it rotates around the axis, it has kinetic energy:
To find the total kinetic energy, we must integrate over the mass:
Noting that v=ωr and that ω is constant for all masses, and so may be taken outside of the integral, the kinetic energy of mass is:
As the term in brackets is the moment of inertia, I, the equation of the form above is derived.