Close

THERMO Spoken Here! ~ J. Pohl © ~ 2017 | www.ThermoSpokenHere.com (A9820-061) |

Motion of a BODY at constrained by forces to follow a circular path is kinematicly described by the radius of the circle and vectors that depict its position
**P**(t), velocity
**V**(t), and
acceleration, **A**(t).

The graphic (right) shows circular geometry described with positive change of angle, θ(t), and angle rate, d(θ(t))/dt set counter-clockwise, arbitrarily. To Earth-bound coordinates, **X**, **Y** (Z-vertical to the page) we assign the vector triad: **I**, **J** (**K** vertical the page). Position of the BODY using these coordinates and basis is written:

P(t)_{BODY}
= r [cosθ(t) I + sinθ(t)J] |
(1)
Position of a BODY restrained to a motion in "circular domain" as a function of time. |

In what follows we will not carry the notation "BODY." The derivative of position, Eqn-1, with respect to time yields the velocity of the BODY (**V(t)**). Next the derivative of that velocity (which is the second derivative of the position) produces a mathematical expression of the acceleration of the BODY (** A (t)**). Below, each is written in a "magnitude times unit vector" form where the unit vectors, **I** and **J** are fixed in direction. The vectors are:

(2)Position |

And the derivative of (2),

(3)Velocity or dP(t)/dt. |

And the derivative of (3),

(4)
Acceleration: d(dP(t)/dt)dt or dV(t)/dt. |

Problems can be solved with the above set of vectors written just as they are. However a simpler, less writing-intensive vector representation is available. We will use the "truth" of the above set (which you can see) to confirm a simpler representing of **P**(t), **V**(t) and **A**(t)..

Once again, notice the above three vectors are written in magnitude-unit vector form. Compare the "direction" parts (the unit vector parts) of the vectors (2), (3), and (4). Immediately we see the direction of **A** to be opposite to the direction of **P**.

Next, using a simple sketch, we see the direction of velocity, **V** is perpendicular to the direction of **P**, and it is directed toward increasing angle, θ. Thus we define **e**_{r}(t), and **e**_{θ}(t).

The graphic representation of the new unit vectors with analytic statements of position, velocity and acceleration vectors are shown (right).

P(t) = r e_{r}(t) |
(5)
Position: r is constant. |

V(t) = r(dθ/dt)e_{θ}(t) |
(6) dθ/dt is the constant rate of rotation. |

A(t) = - r (dθ/dt)^{2} e_{r}(t) |
(7) Here the rotation rate (dθ/dt) is squared. |

Extension of this idea to apply to non-circular motion, that is for a radius that varies in time, r = r(t), follows easily since the radius is a scalar. The new vector basis described here is:

e_{r}(t),
e_{θ}(t) and K, (when needed) |
(8)
The vectors apply to motion in plane. |

The vectors discussed here apply for motion of a body "in a plane" about some point, with constant distance from it. These represent a number of physical cases. To extend the motion to include variation of the radius (but still in a plane) would be the next step of study.