A massless long, narrow tube is fixed to a horizontal plane with a frictionless

Question

A massless long, narrow tube is fixed to a horizontal plane with a frictionless axle. A thin rod of mass M and length L slides without friction inside the tube. (40 points) A. Show that the moment of inertia of the rod about its center of mass is (1/12)"M B. Choose a suitable set of coordinates and write Lagrange's Equations for this system. C. Initially the rod is centered over the pivot and the tube is rotating at an angular velocity of wo. If it is disturbed slightly, what is the speed of the rod along the radial direction as a function of position? (Hints: consider conserved quantities and recall that (x2) because dad dx dt r-ar dt dt D.Findtherad alanda gular speed of the rodafter alongtime lass me thet betsinfinitelylong).

Explanation / Answer

basicaly moment of inertia = I = Mr2

now the distribution has to be considered as the rod which is cylinder shape is intordiced into another as shown in the figure then the inertian will be distributed allon three positive axes = then I = Mr2 / 3

now as the body is three dimensional one then its inertia will further exists along the negative axes aswell therefore individually the moment of inertia will exists along 6 axes , then I = Mr2 / 6

since the cylinder is pivoted at frictionless axel. it implies the impact of axel will also get effected as per above expalination. therefore 6 axes of rod and 6 with respect to the axel puts together an distribution on inertia along 12 directions , therefore I = Mr2 / 12 , since given r = L , then I = ML2 / 12

radial and angular velocities can be obtained by using the relation v= rw where v = linera velocity or radial velocity r = length of the rod = L and w= angular velocity.

since the body rotates with respect to point of axel then , radial velocity = v = distance moved / time taken

here diatance moved is equal to the circunference of the circle = 2x3.14x r = 2x 3.14x L

let t is time taken therefore = v= 2x31.14 xL/ t

now angular velocity ,w = angular diatance / time

here ngular distance = arc length / radius = y / L

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