10.8 Work and Power for Rotational Motion

Rotational motion involves work, energy, and power principles similar to linear motion. The work-energy theorem applies to rotating objects, relating work done by torques to changes in rotational kinetic energy.

Power in rotating systems is the product of torque and angular velocity. These concepts parallel their linear counterparts, with moment of inertia playing a role similar to mass in translational motion.

Work and Power in Rotational Motion

Work-energy theorem in rotational systems

• Work-energy theorem applies to rotational motion relates net work done on a system to change in rotational kinetic energy ($W = \Delta K_{rot}$)
  • $W$ represents net work done on the system by external torques
  • $\Delta K_{rot}$ represents change in rotational kinetic energy of the system (flywheel, gear)
• Rotational kinetic energy depends on moment of inertia $I$ and angular velocity $\omega$ ($K_{rot} = \frac{1}{2}I\omega^2$)
  • Moment of inertia $I$ measures resistance of an object to rotational acceleration depends on mass distribution (solid cylinder, hollow sphere)
  • Angular velocity $\omega$ measures rate of rotation in radians per second
• Work done by a constant torque equals product of torque $\tau$ and angular displacement $\Delta \theta$ ($W = \tau \Delta \theta$)
  • Constant torque $\tau$ applied over an angular displacement $\Delta \theta$ (door hinge, wrench)

Angular velocity from work-energy principles

• Rearrange work-energy theorem to solve for final angular velocity $\omega_f$ ($\omega_f = \sqrt{\frac{2(W + K_{rot,i})}{I}}$)
  • $W$ represents net work done on the system
  • $K_{rot,i}$ represents initial rotational kinetic energy
  • $I$ represents moment of inertia
• Consider initial and final states of the system to determine change in rotational kinetic energy
  • Initial angular velocity $\omega_i$ and final angular velocity $\omega_f$ (spinning up a hard drive, slowing down a fan)

Power in rotating rigid bodies

• Power in rotational motion equals product of torque $\tau$ and angular velocity $\omega$ ($P = \tau \omega$)
  • Torque $\tau$ measures rotational force
  • Angular velocity $\omega$ measures rate of rotation
• Instantaneous power calculated using torque and angular velocity at a specific moment (peak power output of a motor)
• Average power equals work done divided by time interval ($P_{avg} = \frac{W}{\Delta t}$)
  • $W$ represents work done
  • $\Delta t$ represents time interval (power generated by a wind turbine over an hour)

Rotational vs translational work-power equivalents

• Rotational work-energy theorem ($W = \Delta K_{rot}$) analogous to translational work-energy theorem ($W = \Delta K$)
  • Net work changes rotational kinetic energy in rotational systems
  • Net work changes kinetic energy in translational systems
• Rotational kinetic energy ($K_{rot} = \frac{1}{2}I\omega^2$) analogous to translational kinetic energy ($K = \frac{1}{2}mv^2$)
  • Moment of inertia $I$ in rotational systems plays role of mass $m$ in translational systems
  • Angular velocity $\omega$ in rotational systems plays role of velocity $v$ in translational systems
• Work done by a constant torque ($W = \tau \Delta \theta$) analogous to work done by a constant force ($W = F \Delta x$)
  • Torque $\tau$ in rotational systems plays role of force $F$ in translational systems
  • Angular displacement $\Delta \theta$ in rotational systems plays role of linear displacement $\Delta x$ in translational systems
• Power in rotational motion ($P = \tau \omega$) analogous to power in linear motion ($P = Fv$)
  • Torque $\tau$ in rotational systems plays role of force $F$ in translational systems
  • Angular velocity $\omega$ in rotational systems plays role of velocity $v$ in translational systems

Angular Momentum and Rotational Dynamics

• Angular momentum ($L = I\omega$) is conserved in the absence of external torques
  • Conservation of angular momentum applies to systems like figure skaters spinning
• Rotational inertia (moment of inertia) determines an object's resistance to changes in rotational motion
• Angular acceleration measures the rate of change of angular velocity in rotating systems
• Rigid body rotation occurs when all parts of an object rotate about a fixed axis with the same angular velocity