0.29 Phy1260: circular motion -- the mathematics of circular motion  (Page 4/7)

Assume that the arc length is equal to one-half the circumference of the circle. This arc represents a subtended angle of 180 degrees. Then,

s = circumference/2 = pi * radius

angle = (pi * radius)/radius = pi

From this we can see that pi radians is equal to 180 degrees.

Earlier we saw that 2*pi radians equal 360 degrees.

Tangential displacement versus angular displacement

Consider the case of a 1.5-radian angular displacement of a wheel in a given time interval. What is the corresponding displacement of apoint on the circumference of the wheel? Assume that the radius of the wheel is 0.5 meters.

angle = s/r, or

s = angle * r, or

s = 1.5 * 0.5m = 0.75m

A simple solution

Thus, we see that the tangential displacement of a point on the circumference of a wheel due to a given angular displacement of the wheel in radians is simplythe product of the displacement and the radius of the wheel.

Solving for the same result using angular displacement in degrees would be somewhat morecomplicated.

Tangential speed versus angular velocity

A similar simplification occurs when dealing with the angular velocity of a wheel and the tangential speed of a point on the circumference of the wheel.

As in linear measurements, the average angular velocity of a wheel is equal to the angular displacement of the wheel divided by the time interval duringwhich the displacement takes place.

Another example

Consider the case of a 1.5-radian/second angular velocity of a wheel. What is the corresponding tangential speed of a point on the circumference ofthe wheel? Assume that the radius of the wheel is 0.5 meters.

The tangential speed is equal to the tangential displacement, s, divided by the time interval over which the displacement occurs. Given the above information,we can write:

w = s/(r*dT)

Given that

v = s/dT

Substitution yields

v; = w*r, or

v = (1.5/s)*(0.5m) = 0.75 m/s

Another simple relationship

Once again, if you keep your units straight, the tangential speed of a point on the circumference of the wheel is simply equal to the angular velocity inradians per second multiplied by the radius of the wheel.