Physics: Circular Motion

Uniform Circular Motion

Uniform circular motion is defined by a particle or point moving at a constant speed around a circular path.

It is useful to decompose the components of velocity and acceleration into a component tangent to the circular path and perpendicular to the circular path. There is no tangential acceleration if the speed of the circular motion is constant. There is, however, a radial acceleration, $a_"rad"$ ,inward and therefore perpendicular to the circular path. This radial component of acceleration changes the direction of the velocity vector, $vecv$ , but not the speed of the point or particle. /attachments/1f2ebae6-8fb8-11e4-a9fb-bc764e2038f2/Instntaeous velocity - circular motion.png

It an be shown from the ration of the similar sides of the triangles represented in Figure 1 that:

$|Deltavecv| / v_1 = (Deltas)/R$ . And from this directly: $|Deltavecv| = v_1 /R * Deltas$

$|Deltavecv| /(Deltat) = v_1 /R * (Deltas)/(Deltat) = a_"avg"$

As the points get closer together and $Deltas$ gets smaller and smaller, we get the magnitude of the instantaneous acceleration:
$a = lim_(t->0) [v_1/R (Deltas)/(Deltat)] = v_1/R * lim_(t->0) (Deltas)/(Deltat) = v_1/R * v_1$

$a_"rad" = v^2/R$ ^{Centripetal Acceleration ]}

Since the circumference of circular path is circumference = $2*pi*R$ , we can divide this distance by the time it takes to travel around the circumference. The time it takes to travel once around the circumference is the rotational period, T. So, the speed of the point moving in a circular motion is:

$v = (2*pi*R) /T$ ^{Speed of Circular Motion ]}

and since the rotational frequency is related to the velocity and the radius as follows:

$omega = v/R = (2*pi*R) /(T *R) = (2*pi) /T$ [ See the calculator button labeled Angular Frequency from Period ]

Substituting the expression for v into the equation for acceleration, $a_"rad" = v^2/R$ , we get:

$a_"rad" = ((2*pi*R) /T)^2/R = (4 * pi^2*R)/T^2$ ^{Radial Acceleration (R,T) ]}

Non-uniform Circular Motion

For non-uniform circular motion, the tangential acceleration is not zero, as it was for uniform circular motion, so:

$a_"tan" = (d|vecv|) /dt$ ⁴

So, the acceleration vector for non-uniform circular motion is:

$veca = |(dvecv)/ dt| = sqrt(a_"rad"^2 + a_"tan"^2)$ [ See calculator button labeled Acceleration - Non-uniform Circular Motion ]

Physics: Circular Motion

Uniform Circular Motion

Non-uniform Circular Motion

See also