(Solved): 5: The linear mass density of a bar (which stretches from \( x=0 \) to \( x=L \) ) is \( \lambda(x ...

5: The linear mass density of a bar (which stretches from \( x=0 \) to \( x=L \) ) is \( \lambda(x)=\alpha x \). a: Determine \( \alpha \) such that the total mass of the bar is \( 1 \mathrm{~kg} \). \( \mathrm{b} \) : Determine \( \bar{x} \). This is equivalent to finding the center of mass of the bar. c: Determine \( \operatorname{var}(x) \). This is equivalent to finding the moment of inertia of the bar relative to the center of mass.