0.29 Phy1260: circular motion -- the mathematics of circular motion  (Page 5/7)

The relationship between period and frequency

As you already know, when the speed of a point moving in a circle is constant, its motion is called uniform circular motion.

As you also already know, even though the speed of the point is constant, the velocity is not constant. The velocity is constantly changing because the direction of thevelocity vector is constantly changing.

The period

The amount of time required for the point to travel completely around the circle is called the period of the motion.

The frequency

The frequency of the motion, which is the number of revolutions per unit time, is defined as the reciprocal of the period. That is,

frequency in rev per sec = 1/(period in sec per rev), or

f = 1/T

where

• f represents frequency in revolutions per second
• T represents period in seconds per revolution

The relationship between angular velocity and frequency

The speed of a point moving completely around the circle is equal to the distance traveled divided by the time.

sT = 2*pi*r/T, or

sT = 2*pi*r*f

where

• sT is the tangential speed
• r is the radius
• T is the time required for the point to make one complete revolution
• f is the reciprocal of T

We know from before that

sT = w * r, or

w = sT/r

Therefore, by substitution from above,

w = 2*pi*r*f/r = 2*pi*f, or

the angular velocity in radians per second is the product of 2*pi and the frequency in revolutions per second.

where

• sT is tangential speed
• w is angular velocity in radians per second
• f is frequency in revolutions per second, or cycles per second, or hertz

The SI unit for frequency

The SI unit for frequency is hertz (Hz) where 1 Hz is equal to one revolution per second or one cycle per second.

Radial (centripetal) acceleration

In an earlier module, you learned how to subtract vectors and; demonstrate that the acceleration vector of an object moving with uniformcircular motion always points toward the center of the circle. However, in that lesson, we did not address the magnitude of the acceleration vector. We will dothat here.

A very difficult derivation

Deriving the magnitude of the acceleration vector depends very heavily on the use of vector diagrams, complex assumptions, complicated equations.Unfortunately, this is one of those times that I won't be able to present thatderivation in a format that is accessible for blind students. In this case, blind students will simply have to accept the final results in equation form anduse those equations for the solution of problems in this area.