In this section we will use our new definitions for rotational variables to
generate kinematic equations for rotational motion. In addition, we will
examine the vector nature of rotational variables and,
finally, relate linear and angular variables.
Kinematic Equations
Because our equations defining rotational and translational variables are
mathematically equivalent, we can simply substitute our rotational variables
into the kinematic equations we have already derived for translational
variables. We could go through the formal derivation of these equations, but
they would be the same as those derived in OneDimensional
Kinematics. Thus we can simply state the
equations, alongside their translational analogues:
v_{f} = v_{o} + at   σ_{f} = σ_{o} + αt 

x_{f} = x_{o} + v_{o}t + at^{2}   μ_{f} = μ_{o} + σ_{o}t + αt^{2} 

v_{f}^{2} = v_{o}^{2} + 2ax   σ_{f}^{2} = σ_{o}^{2} +2αμ 

x = (v_{o} + v_{f})t   μ = (σ_{o} + σ_{f})t 

These equations for rotational motion are used identically as the corollary
equations for translational motion. In addition, like translational motion,
these equations are only valid when the acceleration,
α, is constant.
These equations are frequently used and form the basis for the study of
rotational motion.
Relationships Between Rotational and Translational Variables
Now that we have established both equations for our variables and kinematic
equations relating them, we can also relate our rotational variables to
translational variables. This can sometimes be confusing. It is easy to think
that because a particle is engaged in rotational motion, it is not also defined
by translational variables. Simply remind yourself that no matter what path a
given particle is traveling in, it always has a position, velocity and
acceleration. The rotational variables we generated do not substitute for
these traditional variables; instead, they simplify calculations involving
rotational motion. Thus we can relate our rotational and translational
variables.
Translational and Angular Displacement
Recall from our definition of angular
displacement
that:
μ = s/r
Implying that
Thus the displacement,
s, of a particle in rotational motion is given by the
angular displacement multiplied by the radius of the particle from the axis of
rotation. We can differentiate both sides of the equation with respect to time:
=
Thus
Translational and Angular Velocity
Just as linear displacement is equal to angular displacement times the radius,
linear velocity is equal to angular velocity times the radius. We can relate
α and a, by the same method we used before: differentiating with respect
to time.
=
r
Translational and Angular Acceleration
We must be careful in relating translationa and angular acceleration because
only gives us the change in velocity with respect to time in the
tangential direction. We know from
Dynamics that any particle
traveling in a circle experiences a radial force equal to . We
must therefore generate two different expressions for the linear acceleration of
a particle in rotational motion:
a_{T}  =  αr 

a_{R}  =  

 =  σ^{2}r 

These two equations may seem a bit confusing, so we shall examine them closely.
Consider a particle moving around a circle with a constant speed. The rate at
which the particle makes a revolution about the axis is constant, so
α = 0
and
a_{T} = 0. However, the particle is being constantly accelerated towards the
center of the circle, so
a_{R} is nonzero, and varies with the square of the
angular velocity of the particle.