Angular position, velocity and acceleration Knowledge-Based Study

Study notes

1. Rotational Kinematics: Angular Position and Displacement

   In rotational motion, the position of a rotating rigid body is defined using an angular coordinate rather than a linear Cartesian coordinate. Consider a reference line drawn from the axis of rotation to a point on the body. The angular position, denoted by $\theta$, is the angle between this reference line and a fixed reference axis (usually the positive x-axis).

   Angular position is measured in radians (rad). One radian is defined as the angle subtended at the center of a circle by an arc equal in length to the radius of the circle:

   $$\theta = \frac{s}{r}$$

   where $s$ is the arc length and $r$ is the radius of the circular path. A full circle corresponds to $2\pi$ radians (360 degrees).

   When a body rotates from an initial position $\theta_1$ to a final position $\theta_2$, its angular displacement $\Delta\theta$ is given by:

   $$\Delta\theta = \theta_2 - \theta_1$$

   By convention, counterclockwise rotation is designated as positive, whereas clockwise rotation is designated as negative.

   Describing the Rate of Rotation: Angular Velocity

   Just as linear velocity describes the rate of change of linear position, angular velocity describes how rapidly an object rotates about its axis.

   The average angular velocity, $\omega_{\text{avg}}$, over a time interval $\Delta t$ is defined as:

   $$\omega_{\text{avg}} = \frac{\Delta\theta}{\Delta t} = \frac{\theta_2 - \theta_1}{t_2 - t_1}$$

   The instantaneous angular velocity, $\omega$, is the limit of the average angular velocity as the time interval approaches zero:

   $$\omega = \lim_{\Delta t \to 0} \frac{\Delta\theta}{\Delta t} = \frac{d\theta}{dt}$$

   Key characteristics of angular velocity include:

   • SI Unit: Radians per second ($\text{rad/s}$).
   • Direction: Angular velocity is a vector quantity along the axis of rotation, with its direction determined by the right-hand rule. For planar motion, we treat counterclockwise rotation as positive $\omega$ and clockwise rotation as negative $\omega$.
   • Period and Frequency: For uniform circular motion, angular velocity relates to the period $T$ and rotational frequency $f$ via $\omega = 2\pi f = \frac{2\pi}{T}$.

   Changing Rotational Speed: Angular Acceleration

   When the angular velocity of a rotating object changes over time, the object undergoes angular acceleration, denoted by $\alpha$.

   The average angular acceleration, $\alpha_{\text{avg}}$, over an interval $\Delta t$ is:

   $$\alpha_{\text{avg}} = \frac{\Delta\omega}{\Delta t} = \frac{\omega_2 - \omega_1}{t_2 - t_1}$$

   The instantaneous angular acceleration, $\alpha$, is defined as:

   $$\alpha = \lim_{\Delta t \to 0} \frac{\Delta\omega}{\Delta t} = \frac{d\omega}{dt} = \frac{d^2\theta}{dt^2}$$

   Key features of angular acceleration include:

   • SI Unit: Radians per second squared ($\text{rad/s}^2$).
   • Speeding Up vs. Slowing Down: If $\omega$ and $\alpha$ have the same algebraic sign, the rotation is speeding up. If $\omega$ and $\alpha$ have opposite signs, the rotation is slowing down.
   • Constant Angular Acceleration: When $\alpha$ is constant, rotational kinematics mirrors linear kinematics directly:
     • $\omega = \omega_0 + \alpha t$
     • $\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2$
     • $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$

   Connecting Linear and Rotational Quantities

   For a point on a rigid body located at a perpendicular distance $r$ from the axis of rotation, its linear motion along the circular arc is directly linked to the angular variables of the body:

   • Linear Distance (Arc Length): $$s = r\theta$$ (where $\theta$ must be in radians)
   • Tangential Speed: $$v_t = \frac{ds}{dt} = r\frac{d\theta}{dt} = r\omega$$ Every point on a rigid body shares the same angular velocity $\omega$, but points further from the axis (larger $r$) have a greater tangential speed.
   • Tangential Acceleration: $$a_t = \frac{dv_t}{dt} = r\alpha$$ This component accounts for changes in the magnitude of linear velocity.
   • Radial (Centripetal) Acceleration: $$a_c = \frac{v_t^2}{r} = \omega^2 r$$ This component points toward the center of curvature and accounts for changes in the direction of the linear velocity vector.

   The total linear acceleration of the point is the vector sum of tangential and radial components: $a = \sqrt{a_t^2 + a_c^2}$.