Consider a mass-and-spring system containing two masses m1=1and m2=1 whose displacement functions x(t) and y(t) satisfy the differential equations below. (a) Describe the two fundamental modes of free oscillation of the system. (b) Assume that the two masses start in motion with the initial conditions x(0)=21 x′(0)=4, and y(0)=33, y′(0)=2 and are acted on by the same force F1(t)=F2(t)=−1080cos(7t). Describe the resulting motion as a superposition of oscillations at three different frequencies. x′′=−7x+6y y′′=9x−22y (a) The lower frequency mode has ω1= .The masses oscillate in ▼ and the amplitude of the oscillation of m2 is times the amplitude of the oscillation of m1.
Simple harmonic motion
Simple harmonic motion is a type of periodic motion in which an object undergoes oscillatory motion. The restoring force exerted by the object exhibiting SHM is proportional to the displacement from the equilibrium position. The force is directed towards the mean position. We see many examples of SHM around us, common ones are the motion of a pendulum, spring and vibration of strings in musical instruments, and so on.
Simple Pendulum
A simple pendulum comprises a heavy mass (called bob) attached to one end of the weightless and flexible string.
Oscillation
In Physics, oscillation means a repetitive motion that happens in a variation with respect to time. There is usually a central value, where the object would be at rest. Additionally, there are two or more positions between which the repetitive motion takes place. In mathematics, oscillations can also be described as vibrations. The most common examples of oscillation that is seen in daily lives include the alternating current (AC) or the motion of a moving pendulum.
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