Crash Course — Unit 5: Torque and Rotational Dynamics

• Rotational Kinematics
  €” Describes how things spin using angular variables, just like linear kinematics described how things move in a line.
• Angular Displacement (Δθ)
  €” The angle (in radians!) through which an object has rotated. It's the rotational version of linear displacement (Δx).
• Angular Velocity (ω)
  €” How fast an object is spinning, measured in radians per second. It's the rotational version of linear velocity (v).
• Angular Acceleration (α)
  €” The rate at which an object's spin speed is changing. It's the rotational version of linear acceleration (a).
• Torque (τ)
  €” A rotational force; a push or pull that causes an object to rotate. No torque, no change in rotation.
• Lever Arm (r)
  €” The distance from the pivot point (axis of rotation) to where a force is applied. Crucial for calculating torque.
• Rotational Inertia (I)
  €” An object's resistance to being spun or having its spin changed. It's the rotational equivalent of mass (m) and depends on both mass and its distribution.
• Rotational Equilibrium
  €” The state where net torque is zero (Στ = 0), so the object's angular velocity is constant. It might be stationary or spinning at a steady rate.
• Newton's Second Law for Rotation (Στ = Iα)
  €” The master equation of the unit. A net torque causes an angular acceleration, just like a net force causes a linear acceleration (ΣF = ma).
• Parallel-Axis Theorem
  €” A tool to find the rotational inertia of an object spinning around an axis that doesn't pass through its center of mass.

Key Formulas / Terms

• Rotational Kinematics (for constant α)
  • ω = ω₀ + αt
  • Δθ = ω₀t + ½αt²
  • ω² = ω₀² + 2αΔθ
• Connecting Linear and Rotational Motion
  • Arc Length: s = rθ
  • Tangential Velocity: v = rω
  • Tangential Acceleration: a_t = rα
• Rotational Dynamics
  • Torque: τ = rFsinθ (where θ is the angle between the lever arm r and the force F)
  • Newton's Second Law for Rotation: Στ = Iα
  • Rotational Inertia (general form): I = Σmr²
  • Parallel-Axis Theorem: I = I_cm + Md²

Exam Traps

• Trap
  Radians vs. Degrees. · Counter: Your calculator is probably in degree mode. For this unit, it must be in RADIAN mode. All the rotational formulas are built on radians. Make it a habit to check your calculator mode before every problem.
• Trap
  Confusing Torque (τ) with Force (F). · Counter: Force is a push or pull. Torque is a twist. A force only creates a torque if it's applied at a distance from a pivot. Always ask, "Where is the pivot?" and "Is this force trying to cause a rotation around it?" before you calculate τ = rFsinθ.
• Trap
  Using the wrong distance for r in the torque equation. · Counter: r is the "lever arm," the specific distance from the pivot point to the exact spot where the force is applied. It is not always the length of the whole object. Identify your pivot first, then find the distance r.
• Trap
  Forgetting sinθ when the force isn't perpendicular. · Counter: If a force is applied at an angle, only the component of the force perpendicular to the lever arm creates torque. The sinθ in τ = rFsinθ isolates that component for you. If the force is perfectly perpendicular, θ = 90° and sin(90°) = 1, so you can ignore it. Otherwise, you must include it.
• Trap
  Treating Rotational Inertia I like a fixed property. · Counter: I is not like mass. It changes depending on the axis of rotation. A baseball bat is harder to swing from the skinny end (high I) than from the fat end (low I). Always check the formula sheet for the correct I for the shape and axis given in the problem.