Question

A mass is connected to the center of a rotating disk via a spring. The mass is constrained by a smooth (frictionless) slot to only move in a purely radial direction X Lo The disk has radius Rand starts rotating from rest at a constant angular acceleration (counterclockwise). The starting position of the mass is as shown, where its distance away from the center of the disk is the same as the resting length of the spring. Gravity points into the page. Determine a differential equation of motion for the mass that describes the movement of the mass as the disk rotates.

Answer 1

Solution L. K Fallowing Ls Hhe maHhe matical model and Solution metHhod. The variable are fallooing the '.- Sliding mass in Hhe slot M spring stiffness the relaxed leng th ( Freeleng H Lo spring current position (length This is the of spring along the radius megsured from mass the -. tha disc centre of Derivation of equation of motion There are two kind of accelaration ü) Along the direction of mass in X slot (i Perpendli cular direction of mass (öx+26 Centre Here we need to determine the equation of motion of osulatin7 macs in the slot , discucs about the perpendiur accelaration So these ic no need

motion of sprina mass consider the oscillatay Let us tem Neston's Method k (x-Lo) 2 M. 9. .2 оо Мx+ k(x-Lo) ме.х where e is angular velocity of disc. + kz -M .2 kLo мех .2 k Lo (k-- Мә ) ми+ Answer This is the differential equation of motion

similarly for disc,if needed determine the equation of motion we Can accelaration (angular mass of the disc is M is then the equation of motion of disc I e + 2 M XO Y where I is Hhe moment of Inertia about centre SSbuA 2. 2 Mi. Rt Mx putting this aloove 2 - 2m e X A 2 7MRMx2 This is equation of motion fa disc