8.2 Uniform Circular Motion

The kinematics of uniform circular motion were introduced in 《 SECTIONS 4.4–4.5, and a review is highly recommended. Now we’re ready to study dynamics—how forces cause circular motion. FIGURE 8.3 reminds you that the particle’s velocity is tangent to the circle, and its acceleration—a centripetal acceleration—points toward the center. If the particle has angular velocity $ω$ and speed $v = ω r$ , its centripetal acceleration is

\vec{a} = (\frac{v^{2}}{r}, toward center of circle) = (ω^{2} r, toward center of circle)

(8.4)

The figure consists of two parts that illustrate the uniform circular of a particle and the r t z coordinate system.

FIGURE 8.3 Full Alternative Text

An xy-coordinate system is not a good choice to analyze circular motion because the x- and y-components of the acceleration are not constant. Instead, as FIGURE 8.3 shows, we’ll use a coordinate system whose axes are defined as follows:

The origin is at the point where the particle is located.
The r-axis (radial axis) points from the particle toward the center of the circle.
The t-axis (tangential axis) is tangent to the circle, pointing in the counterclockwise direction.
The z-axis is perpendicular to the plane of motion.

These three mutually perpendicular axes form the rtz-coordinate system.

You can see that the rtz-components of $\vec{v}$ and $\vec{a}$ are

\begin{array}{l} v_{r} = 0 & a_{r} = \frac{v^{2}}{r} = ω^{2} r \\ v_{t} = ω r & a_{t} = 0 \\ v_{z} = 0 & a_{z} = 0 \end{array}

(8.5)

where the angular velocity $ω = d θ / d t$ must be in rad/s. For uniform circular motion, the velocity vector has only a tangential component and the acceleration vector has only a radial component. Now you can begin to see the advantages of the rtz-coordinate system.

NOTE Recall that $ω$ and $v_{t}$ are positive for a counterclockwise (ccw) rotation, negative for a clockwise (cw) rotation. The particle’s speed is $v = | v_{t} |$ .