A stick of mass 1.3 kg and length 0.7 m has a mass...

A stick of mass 1.3 kg and length 0.7 m has a mass distribution ?(x) = ax where a is a constant and x is measured from its one end. What is the moment of inertia Icm (in kg. m²) of the stick with respect to an axis that is perpendicular to the stick and goes through its center-of-mass? 0.053 0.212 0.106 0.035 0.318

Transcript

00:02 So for this problem we'll be given that there's a stick of mass 1 .3 kilograms and a length of 0 .7 meters and we also will be given that the mass distribution of the stick is lambda of x 8 times x meaning it's proportional to its length okay so x is measured from one end one end of the stick and yeah the the density the mass distribution the mass density increases further away from the from from this specific end of the stick okay how do we define this mass distribution is yeah in one dimension it can be can be defined by dm by dx meaning dm is the infinitesimal infinitesimal mass for corresponding to an infinitesimal displacement dx okay so therefore dm this mass equals density times dx okay here i hope this is clear this is the basic definition of densities or mass distributions and again i'm using m capital for the total mass so and what we want to calculate we want to calculate the moment of inertia of the stick with respect to an axis that is perpendicular to the stick to the body of the stick it goes through the center of mass so if let's say that this is the center of mass of the stick the axis again it will be perpendicular to the body of the stick and it will go through this specific point like this okay okay how do we go about solving this kind of problem the first thing that we do we need to find the proportionality constant of the density of the mass distribution and how do we do this okay it's mass mass density therefore the its integral by definition its integral over the whole length of the stick it should be equal to mass capital okay therefore if we really quickly do the the integral i'm not gonna go in much into that yeah i assume that you can do this kind of integrals yeah the indefinite integral is just x squared over 2 x squared 0 to l equals m yeah and if we do the calculations there we see that uh yeah the constant a equals two times two times m over l squared okay this way we're able we were able to to calculate the proportionality constant then we need to find the x the exact position of the center of mass okay and we'll denote this as x cm meaning it's the center of mass with respect or measured from the from one end of the stick okay and how do we do this let me i'm gonna write the in the formal now formula for the center of mass in one dimension anyway don't be the integrals of x dm over the whole body of the stick over dm body of the stick okay this might look a little bit weird but anyway if we divide this infinitesimal displacement by dx we get immediately the the mass distribution or the mass density and it makes more sense again we're dividing dm by dx using the the chain rule and we get x lambda of x because this is dm by dx and the limits of course are from 0 to l and the same from here of x dx from 0 to l and we immediately recognize that this is basically this integral so the expression of the denominator is just m and all we have to do is calculate this expression here so let's do it really really quickly um i assume that this is not a difficult task to do um yeah the indefinite integral is just x x cubed over 3 from 0 to l and we're getting a l cubed over 3 and if we did and we have already calculated a so that will be 2 2 times m l cube yeah r squared m 3 and we see immediately a simplification there oops what did i do r squared square not squared here okay so what we see and we we mustn't forget that in order to calculate x cm we need to divide by mass so this…

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