PH 212

Image source

Circular motion

Uniform circular motion

Circular motion kinematics correlates directly to linear kinematics. Equations for position, velocity and acceleration are similar in form to equations for angle, angular velocity and angular acceleration.

The left column gives the equations for linear kinematics in one dimension. The equations on the right are the corresponding equations for circular motion.

Theta is defined as an angle in a circle of radius r.

Circular motion graphical representations are similar to those for linear kinematics. The time derivative of q gives the angular velocity w The time derivative of w gives the angular acceleration a w gives the rate of change of the angle. It has units of radians per second, so can also be viewed as a frequency. a is the rate of change of w.

You may also need to express circular motion of an object or a point. To do that, we can define velocity and acceleration using the arclength formula. Here, we have defined the tangential acceleration at and the radial acceleration ar. To completely define acceleration in three dimensions, we can include the acceleration in the z-direction az.

 

For uniform circular motion, we can relate the tangential velocity v to the period T.

The acceleration vector for circular motion, in polar coordinates, is mathematically equivalent to the acceleration vector in Cartesian coordinates. In both cases, the components are independent, in that they are perpendicular to each other. Acceleration in polar coordinates is used for rotating systems as a convenience.

 

It is important to note that these linear accelerations are two representations of acceleration, differing in coordinate system only. The acceleration of an object can be defined by either form. We choose the form that is most convenient for the motion of the object in question.

 

Angular acceleration, on the other hand, is a different kind of quantity. It defines the motion of rotation, not of translation.

Let's review vectors a bit. Which way does the resultant vector v2 - v1 point?

Consider the diagram below. Which direction does v2 - v1 point? How about v3 - v2? How about v4 - v3?

All Dv vectors point radially inward. We can extend this concept to see that as we increase the number of vectors and approach a circle, the Dv vectors continue to point radially inward.

Derive radial acceleration

 

For uniform circular motion, the magnitudes of the velocity vectors are equal: v1 = v2. The velocity vectors are at right angles to the radial vectors, so the triangle made by the velocity vectors is similar (in the geometric sense) to the triangle made by the radial vectors.

 

These similar triangles provide a method for deriving a convenient form of the radial acceleration. For circular motion, the direction of the radial acceleration always points toward the center of the circle.