Consider the gibbs vector defined as where g = (g 1.8283), e = (el, e2. e3) is euler's eigenaxis vector, and θ is the angle associated with euler's eigenaxis rotation. (a) show that the kinematic differential equation for the gibbs vector can be found as gi 82 | =う! g281 83 1+ 8182-83 g183+82][en 83 88283-8i«i 83 8 8382g 18 on or where 0 -83 82 0 81 82 8 0 (b) also show that for infinitesimal rotations, we have

Aball has an initial velocity of 3 m/s. if there is no friction, what is the highest it could roll?

Atelevision set that has been running for a while heats up even the air around it. which two laws of thermodynamics could be used to analyze this scenario

Consider a uniformly charged ring in the xy plane, centered at the origin. the ring has radius a and positive charge qdistributed evenly along its circumference. a)what is the direction of the electric field at any point on the z axis? . b)what is the magnitude of the electric field along the positive z axis? use k in your answer, where k=14πϵ0. d)the ball will oscillate along the z axis between z=d and z=−d in simple harmonic motion. what will be the angular frequency ω of these oscillations? use the approximation d≪a to simplify your calculation; that is, assume that d2+a2≈a2. express your answer in terms of given charges, dimensions, and constants