Question

A homogeneous ring of radius R and mass m can roll on a horizontal surface without slipping. It is attached at the center to a spring of elastic constant k and rest length 1, and can oscillate on the horizontal plane. See the figure below for a schematic presentation. k (i) What is the number of degrees of freedom of the system? [2] (ii) Compute the moment of inertia of the ring about an axis perpendicular to it and going through its center of mass. [5] (iii) Write down the Lagrangian of the system and the Euler-Lagrange equation. [12] (iv) Find the general solution of the Euler-Lagrange equation and the frequency of small oscillations. [6]

Answer 1

Solution

i) One. The system has only one degree of freedom.

ii) iii) and iv) see images.

C) Solution do Its centar ode=Rao PAR 1) Moment of inertia of sing of radius R and mass m through ais axis passing through and I to its plane, I Sredm an element of sing covered by angle del. Its mass dma trade is the mass density of ring. Also the perpendicular for same (= 29 distance mass element dan. every 207 Thus, 29 R2 CARDO) - R²1 Idee Consider where 27 R to the aods 63 22 DR3 But dam IP zak JoB2 2XP of spring. as the - I = MR². in considering & as the generelized coordinate, k Kinetic energy of system Rotational kinetic energy of sing. I w2 T - 2 m R² & 2 1 - mp3 and war 0} Potential energy of system Potential energy Al-R6 it will extend If l is the rest length of spring sing rotates & radians (sire the song is solling without slopping). This spring force P., &- 3 k ( (l+Al) – e) 2 k () { Bl=R83. Thus Lagrangian of system, L=T-N L - imp?82-k Rºp? Faler -Lagrange equation, ( 8 ) - 24 (mR24) - FKRP) 0 mR? - 2x 1 KR? a) mp24 + kR²6=0 - 1612²p ko DL op - 2x4 me? 4 d dt ƏL 24 et