Two equal masses on an effectively friction less horizontal air track are held between rigid supports by three identical springs, as shown. The displacements from equilibrium along the line of the springs are described by coordinates xA and xn, as shown. If either of the masses is clamped, the period T(=2 π/ ω) for one complete vibration of the other is 3 sec. (a) If both masses are free, what are the periods of the two normal modes of the system? Sketch graphs of xA and xB versus t in each mode
Two equal masses on an effectively friction less horizontal air track ar…
Jul 1, 2021
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SOLVED: The system shown has mass m1 and mass m2, held by 3 springs each of stiffness k. Initially, it is in a static equilibrium position. Mass m is displaced 1 inch and then released. a. Obtain the two equations of motion for x1(t) and x2(t).b. Assume periodic motion of various amplitudes and frequencies: X1 = Acos(wt) and X2 = Bcos(wt). Find the characteristic equation.c. Solve the frequency equation to get natural frequencies w1 and w2.d. Find amplitude ratios (A/B1 and A/B02).e. State the 4
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If a mass m is placed at the end of a spring, and if the mass is pulled downward and released, the mass-spring system will begin to oscillate. The displacement y of the mass from its resting position is given by a function of the form y(t)=c1 cosωt+c2 sinωt where ωis a constant that depends on the spring and the mass. (See the figure below.) Show that the set of all functions described in (5) (with ωfixed and c1, c2 arbitrary) is a vector space. | Numerade
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