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Rotation & Angular Dynamics · Physics

Work and power have clean rotational forms.

A torque can do work when it turns an object through an angle. Power in rotation links torque and angular speed, making it easy to analyze motors, pulleys, and energy transfer.

This topic

Work and Power in Rotational Motion

Connect torque to work, interpret rotational power, and use energy methods when forces are complicated.

Work
Torque doing work
Work in rotation is the energy transfer associated with torque producing angular displacement. The same “force along displacement” idea appears in an angular form.
  • Work from torque for a fixed axis
  • Sign of work: speeding up vs slowing down
  • When torque does zero work (no rotation)
  • Units check: N·m is a joule (work), but torque is not energy
Start nano-lesson Work needs an angle turned.
Formula
Rotational work formula (constant torque)
For constant torque about a fixed axis, work equals torque times angular displacement. This mirrors W = Fd for constant force along a line.
  • W = τΔθ for constant τ (about a chosen axis)
  • Δθ must be in radians
  • Interpreting the sign of τ and Δθ
  • Connecting to ΔKrot via work–energy
Start nano-lesson Radians make the units work out.
Power
Rotational power
Power measures the rate of doing work. In rotational systems, power connects directly to torque and angular speed, which is especially useful for motors.
  • P = dW/dt in general
  • For rotation: P = τω (instantaneous)
  • Why high ω can mean high power even with modest τ
  • Interpreting motor specs (conceptually)
Start nano-lesson Power links torque to speed.
Strategy
Energy methods vs torque dynamics
Sometimes torque analysis requires unknown forces (like friction or tension). Energy methods can bypass those forces if you track work done and energy changes correctly.
  • When energy is easier than Στ = Iα
  • External work vs internal constraints
  • Choosing the system and identifying energy forms
  • Common pitfalls: wrong sign, wrong angle unit
Start nano-lesson Pick the method that avoids unknowns.
Practice
Practice & Exercises
Practice computing rotational work and power, and solving for ω or τ using energy and rate ideas.
  • Compute W from τ and Δθ (radians)
  • Use work–energy: W = ΔKrot
  • Use P = τω for motors and pulleys
  • Mixed translation + rotation energy problems
  • Exam-style work/power in rotation sets
Open practice sets Always convert degrees to radians first.