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

The frequency of vibrations of a vibrating violin string is given by f = 1 2l t ρ where l is the length of the string, t is its tension, and ρ is its linear density.† (a) find the rate of change of the frequency with respect to the following. (i) the length (when t and ρ are constant) (ii) the tension (when l and ρ are constant) (iii) the linear density (when l and t are constant) (b) the pitch of a note (how high or low the note sounds) is determined by the frequency f. (the higher the frequency, the higher the pitch.) use the signs of the derivatives in part (a) to determine what happens to the pitch of a note for the following. (i) when the effective length of a string is decreased by placing a finger on the string so a shorter portion of the string vibrates df dl 0 and l is ⇒ f is ⇒ (ii) when the tension is increased by turning a tuning peg df dt 0 and t is ⇒ f is ⇒ (iii) when the linear density is increased by switching to another string df dρ 0 and ρ is ⇒ f is ⇒