Question

5. 10 points Two ideal massless springs with spring constant k are connected to two masses that hang vertically as shown in the figure. The top one has mass 3m and the bottom one has mass 2m. The system is only able to oscillate in the vertical direction. a) Determine the equations of motion. (4 pts) b) Find the frequencies of the normal modes of this system for small vertical dis- placements. (4 pts) c) Describe the relative motion and amplitudes of each of the normal modes. (2 pts) 000000000000000 2m)

Answer 1

heleelle elules mass 2 T = 2 Solution - Griven two ideal mossless spring with spring constant k are connected two masses that hong vertically a). Let x, and X, be displacement from mean Iz l Position Kinetic energy of having Jist { (3m) i Kinetic energy of having and T = + (2m) x² Total K T = Ź (2 m) X £ (3m) Čz Potential energy (V)= {K(1,-0j? + (1,-2,)? v=ť [kx?]+ 1 / x L x ² + 1 - 2 x, X2] V: ** [2.'+ 27,' -23,2.] mass → 2 2 2 0 at À = 17 v[N བ 3m 2 고 - K normal k 2 mode. for frequency of det 10-w' fl=0 O Page 1

Thus the T-V DL OL d dt 1 = 0 a je, ali =0 for given egn. egn of motion is :- lagrangian L = 1 (2m) % ² + 1 (3m) 1,² - 1 k[ x, + 2x7 - 2x, 22) egn of motion for a : in et [ 2m3 ] + K2, -KZ, 2 má + kr,-key = 0 2m är tk (x,-X2) = 0 egn. of motion for de [ 24 ] [ 3mi;] 3 miz] + 2i X, + 2K X2 -kx, :0 3 m%; + 2kX, -kx, =0 X2 әL =0 olz dr 3m x₂ + k/2x₂-x)=0 the These are mode equations b). For normal the values of 7 and Ons of motion frequencies. Putting eyni 0 [Page - 2 고 in we get

2. 스m 쓸 2 det k - 3맨 일 2 2 트 2. k-3n ( ) - k W? h 2. 2. 2. k 키 +3m W 3mk 2 m 4 20 2 2 2. 7mw k K 3y + Q 시 ។ ។ 1 + 를 20 솓 2 We . 2 Quadratic 9m in are s- w² 3 2 roots mkt 큐께 니오 m 100 - 3m2 Fau 큐끼 27 잎 25 22 mk 의 3m2 2 5 m크유개 3m2 : 12mk 2조 h 3m2 3m2 K Wa K m LEA-3 6m

is simply the c). Relative motion ! → A collision between two bodies can always be described in a frame of reference in which the total momentum is zero. si The amplitude the Amplitude ! → maxi mun displacement of the object from the equilibrium position. So the same equation applies to the position Simple harmonic motion and dimension of the the position of an object uniform circular motion . Pag-4 e xperiencing object of on one experiencing