Torque and Angular Momentum “Twisty-ness” and other rotational analogues

Review of linear and rotational analogues x, y, z – position v – velocity a – acceleration Kinematic equations: 𝑣= ∆𝑥 ∆𝑡 𝑎= ∆𝑣 ∆𝑡 ∆𝑥= 1 2 𝑎 𝑡 2 + 𝑣 0 𝑡 θ – angle ω – angular velocity α – angular acceleration Kinematic equations: 𝜔= ∆𝜃 ∆𝑡 𝛼= ∆𝜔 ∆𝑡 ∆𝜃= 1 2 𝛼 𝑡 2 + 𝜔 0 𝑡 Linear ↔ Rotational 𝑠=𝑟𝜃 𝑣=𝑟𝜔 𝑎=𝑟𝛼

Preview of linear and rotational analogues F – force m – mass p - momentum τ – torque I – moment of inertia L – angular momentum

Torque A rotating force; influence that causes changes in the rotational motion of an object. 𝝉=𝒓×𝑭=𝑭 𝒓 𝐬𝐢𝐧 𝜽

Example A bolt is loosened when a 30.0 N force is applied perpendicularly to a 5.0 cm wrench. What is the torque on the bolt? [Answer in units of Nm.]

Example What force is necessary to loosen the same bolt if the force is applied at a 45°(π/4) degree angle from the wrench?

Assignment Complete the problems on the ½ sheet of paper. Be ready to white board any one of them tomorrow. (That means do all 7.)

Moment of Inertia Rotational analog of mass - varies by object and rotational axis. Generally: 𝐼= 𝑛 𝑚 𝑛 𝑟 𝑛 2 = 𝑚 1 𝑟 1 2 + 𝑚 2 𝑟 2 2 +… 0 𝑀 𝑟 2 𝑑𝑚 Point mass: 𝑰=𝒎 𝒓 𝟐

I = moment of inertia R = radius L = length M = mass Moment of Inertia

Angular momentum Rotational analog of linear momentum (p = mv) 𝐿=𝑟×𝑝=𝑚 𝑣 𝑟 sin 𝜃 Kepler’s 2nd Law The product of the moment of inertia and the angular velocity 𝑳=𝑰×𝝎 Conserved if there is no external torque on the object A vector quantity

Angular momentum To determine direction, use the right hand rule: Curl fingers in direction of rotation Thumb points in direction of L. http://www.wimp.com/demonstrationmomentum/

Assignment Create a double bubble map to compare linear and rotational motions. You need 2-3 colors.