Relating linear and angular quantities Knowledge-Based Study

Study notes

1. Connecting Arc Length to Angle

   When an object rotates about a fixed axis, every point on the object moves along a circular path. The linear distance traveled along this path is the arc length, denoted by s.

   By definition, the angular displacement $\theta$ measured in radians is the ratio of arc length to the radial distance $r$ from the rotation axis:

   $s = r\theta$

   Key points to remember:

   • The angle $\theta$ must always be expressed in radians (rad), not degrees.
   • Points farther from the axis (larger $r$) trace out a greater arc length $s$ for the exact same angular displacement $\theta$.

   Linear Speed on a Rotating Body

   Differentiating arc length with respect to time yields the relationship between linear speed along the circular path (tangential speed, $v_t$) and angular velocity ($\omega$):

   $v_t = \frac{ds}{dt} = r \frac{d\theta}{dt} = r\omega$

   Implications for rigid body rotation:

   • Every point on a rigid rotating body shares the exact same angular velocity $\omega$.
   • The tangential speed $v_t$ is directly proportional to the radius $r$. A point on the outer rim of a disk moves with a higher linear speed than a point near the center.

   Components of Linear Acceleration

   A particle undergoing circular motion can experience two perpendicular components of linear acceleration:

   • Tangential Acceleration ($a_t$): Represents the rate of change of tangential speed. Related to angular acceleration $\alpha$ by:
     $a_t = \frac{dv_t}{dt} = r\alpha$
   • Radial / Centripetal Acceleration ($a_c$): Directs toward the center of curvature and accounts for the changing direction of the velocity vector:
     $a_c = \frac{v_t^2}{r} = r\omega^2$

   The magnitude of the total linear acceleration $a$ is given by the Pythagorean sum of these mutually perpendicular vectors:

   $a = \sqrt{a_t^2 + a_c^2} = \sqrt{(r\alpha)^2 + (r\omega^2)^2}$

   Overview of Relationships

   For any point at distance $r$ from the axis of rotation:

   • Displacement: $s = r\theta$
   • Velocity: $v_t = r\omega$
   • Tangential Acceleration: $a_t = r\alpha$
   • Centripetal Acceleration: $a_c = r\omega^2$

   Note on Units: Since radians are dimensionless, multiplying an angular quantity (rad, rad/s, rad/s2) by a radius in meters directly produces the corresponding SI linear quantity (m, m/s, m/s2).