Question

SHM worksheet 2 To determine the angular frequency constant o in SHM one can analyze the forces on the oscillating system and use Newton's 2 law to determine the relationship hetwen position and the acceleration. Another technique is to determine the total system. Since we are considering only cases that obey the energy cons it must be true that dE/dt-0. The problems below are easier to do with the energy method energy of the servation principle, A spring of constant k is connected to cylinder (1-MR /2) rolling on a horizontal plane. Show that oo (2k/3M 2K A) Using Newton's force formulation Fama B) Using Newton's torque formulation C) Using Energy conservation dt ?? 2
Please show all work and ignore the work done

Answer 1

Using Energy conservation E = 1/2 Mv^2 + 1/2 I (d theta/dt)^2 + 1/2 kx^2 Now d theta/dt = v/R & I = 1/2 MR^2 therefore E = 1/2 MV^2 + 1/2 MR^2 (V/R)^2 + 1/2 kx^2 = 3/4 MV^2 + 1/2 kx^2 Now dE/dt = 0 implies 0 = 3/4 2 M V dv/dt + 1/2 k 2 x dx/dt ie v (3/2 M d^2 x/dt^2 + kx) = 0 or d^2 x/dt^2 + 2k/3M x = 0 General eq^n is d^2 x/dt^2 + w^2 x = 0 comparing w^2 = 2k/3m implies w = squareroot 2k/3M