Rotational Dynamics

Translational-Rotational Analogues & Connections Continue! Translation Rotation Displacement x θ Velocity v ω Acceleration a α Force (Torque) F τ Mass m ? CONNECTIONS v = rω atan= rα aR = (v2/r) = ω2 r τ = rF

Analogous to Newton’s 2nd Law: Goal: Newton’s 2nd Law for rotational motion. We’ve just seen: The angular acceleration is  the net torque or α  τnet = ∑τ = sum of torques Analogous to Newton’s 2nd Law: a  Fnet = ∑F = sum of forces Recall, Newton’s 2nd Law: ∑F = ma. a  α , F  τ Question: What plays the role of mass m for rotational problems?

Simplest Possible Case A mass m moving in a circle of radius r, one force F TANGENTIAL to the circle τ = rF Newton’s 2nd Law + relation (a = rα) between tangential & angular accelerations  F = ma = mrα So τ = mr2α  Newton’s 2nd Law for Rotations Proportionality constant between τ & α is mr2 (point mass only!)

 Moment of Inertia of the body  For one particle, moving in a circle, Newton’s 2nd Law for rotational motion is: τ = mr2α Extrapolate to a rigid body, made of many particles. Newton’s 2nd Law, rotational motion becomes ∑τ = ∑(mr2)α Since α is the same for all points in the body, it comes out of the sum.  Write ∑τ = Iα, where the proportionality constant I = ∑(mr2) = m1r12 + m2r22 + m3r32 + …  Moment of Inertia of the body sum over all point masses in the body  

Moment of Inertia of the body:  Newton’s 2nd Law, rotational motion is: The net torque = (moment of inertia)  (angular acceleration) Moment of Inertia of the body: Moment of Inertia I = A measure of the rotational inertia of the body. Analogous to the mass m = A measure of translational inertia of the body.

Translational-Rotational Analogues & Connections Translation Rotation Displacement x θ Velocity v ω Acceleration a α Force (Torque) F τ Mass (moment of inertia) m I CONNECTIONS v = rω, atan= rα, aR = (v2/r) = ω2r τ = rF , I = ∑(mr2) NEWTON’S 2nd LAW: ∑F = ma; ∑τ = Iα

Obtained for simple shaped bodies using calculus. Useful for problem solving!

Idisk > Irod , even if their masses are the same!!! Moment of Inertia: I = ∑(mr2) Depends on mass m AND on the distribution of mass (r2) in the object! Idisk > Irod , even if their masses are the same!!!   Disk or thin cylinder   Long, small radius cylinder

I depends on the rotation Example I = ∑(mr2) a) I = (5)(2)2 +(7)(2)2 = 48 kg m2 b) I = (5)(0.5)2 + (7)(4.5)2 = 143 kg m2 NOTE! I depends on the rotation axis and on the mass distribution