10.3 Relating Angular and Translational Quantities

Rotational motion mirrors linear motion, with angular quantities replacing their linear counterparts. Angular position, velocity, and acceleration describe circular movement, while equations for rotational motion parallel those for linear motion.

Understanding rotational dynamics is crucial for analyzing spinning objects. Concepts like torque, moment of inertia, and angular momentum help explain how objects rotate and maintain their spin, similar to force, mass, and linear momentum in linear motion.

Rotational Motion and Angular Quantities

Translation of kinematic equations

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• Linear position ($x$) translates to angular position ($\theta$) in rotational motion
• Linear velocity ($v$) translates to angular velocity ($\omega$) which measures the rate of change of angular position
• Linear acceleration ($a$) translates to angular acceleration ($\alpha$) which measures the rate of change of angular velocity
• Equations for rotational motion mirror those for linear motion:
  • $\omega = \frac{d\theta}{dt}$ relates angular velocity to the rate of change of angular position similar to $v = \frac{dx}{dt}$ in linear motion
  • $\alpha = \frac{d\omega}{dt}$ relates angular acceleration to the rate of change of angular velocity similar to $a = \frac{dv}{dt}$ in linear motion
  • $\theta = \theta_0 + \omega_0t + \frac{1}{2}\alpha t^2$ describes angular position as a function of initial angular position ($\theta_0$), initial angular velocity ($\omega_0$), angular acceleration ($\alpha$), and time ($t$) similar to $x = x_0 + v_0t + \frac{1}{2}at^2$ in linear motion
  • $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$ relates final angular velocity ($\omega$) to initial angular velocity ($\omega_0$), angular acceleration ($\alpha$), and change in angular position ($\theta - \theta_0$) similar to $v^2 = v_0^2 + 2a(x - x_0)$ in linear motion

Calculation of linear quantities

• Linear distance traveled by a point on a rotating object can be calculated using $s = r\theta$ where $r$ is the distance from the point to the axis of rotation and $\theta$ is the angular displacement (in radians)
• Linear velocity of a point on a rotating object can be calculated using $v = r\omega$ where $r$ is the distance from the point to the axis of rotation and $\omega$ is the angular velocity (in radians per second)
  • The direction of linear velocity is always tangent to the circular path traced by the point
• Linear acceleration of a point on a rotating object has two components:
  • Tangential acceleration $a_t = r\alpha$ where $r$ is the distance from the point to the axis of rotation and $\alpha$ is the angular acceleration (in radians per second squared)
  • Centripetal acceleration $a_c = \frac{v^2}{r} = r\omega^2$ where $v$ is the linear velocity, $r$ is the distance from the point to the axis of rotation, and $\omega$ is the angular velocity (in radians per second)
    • The direction of centripetal acceleration is always toward the center of the circular path traced by the point (perpendicular to the linear velocity)

Relation of accelerations to angular motion

• Tangential acceleration ($a_t$) in circular motion:
  • Results from angular acceleration ($\alpha$) which changes the magnitude of the angular velocity and consequently the linear velocity
  • Calculated using $a_t = r\alpha$ where $r$ is the distance from the point to the axis of rotation and $\alpha$ is the angular acceleration
• Centripetal acceleration ($a_c$) in circular motion:
  • Results from uniform circular motion where the magnitude of the linear velocity remains constant but its direction changes continuously
  • Calculated using $a_c = \frac{v^2}{r} = r\omega^2$ where $v$ is the linear velocity, $r$ is the distance from the point to the axis of rotation, and $\omega$ is the angular velocity
• The total acceleration of a point in circular motion is the vector sum of tangential and centripetal accelerations given by $a = \sqrt{a_t^2 + a_c^2}$
  • The direction of the total acceleration depends on the relative magnitudes of tangential and centripetal accelerations (e.g., if $a_t = 0$, the total acceleration points toward the center of the circular path; if $a_c = 0$, the total acceleration is tangent to the circular path)

Rotational Dynamics and Energy

• Torque is the rotational analog of force, causing angular acceleration
• Moment of inertia represents an object's resistance to rotational acceleration, similar to mass in linear motion
• Angular momentum is conserved in the absence of external torques, analogous to linear momentum
• Rotational energy is the kinetic energy associated with an object's rotation
• The parallel axis theorem relates the moment of inertia about any axis to the moment of inertia about an axis through the center of mass