12-g3of the mass m. The spring, pulley, and ropeshave negligible mass.Find the period of vertical oscillationskmProblem 12-g3.

let assume that a body is pullimg the pulley downward by a distance x, then the block moves downward by a distance 2xTotal length of the rope is constantSo, if dxdt=v is the velocity of the pulley, then, 2dxdt=2v is the velocity of the blockSo, the kinetic energy of the block will beK.E=12m2v2=2 mv2Potential energy of the spring will be,P.E=12Kx2Total energy of a simple harmoic motion is constant.then, K.E+P.E=Constant⇒2mv2+12Kx2=constantDifferentiating the equation with respect to time⇒4mvdvdt+12×2Kx×dxdt=0⇒4mvdvdt+Kxv=0⇒4mdvdt+Kx=0 Equation 1acceleration of the pully, aP=dvdtso, acceleration of the block ab=d2vdthence, put aP=dvdt⇒ab=2 aP⇒aP=ab22put the values of aP=dvdtin equation 1⇒4maP+kx=0put, aP=ab2 in above equation⇒2mab=-kx⇒ab=-k2mxwe can write, ab=d2vdt=d22xdt2⇒d22xdt2=-k2mx=-k4m2x where the angular frequency, ω=k4m=-ω22xSo, the time period T=2πω⇒T=2π4mkAnswer: Period of vertical oscillation of mass m is 2π4mk

Answered: 12-g3 of the mass m. The spring,…

12-g3 of the mass m. The spring, pulley, and ropes have negligible mass. Find the period of vertical oscillations k m Problem 12-g3.