An object of mass 3 grams is attached to a vertical spring with spring constant 27 grams/secʻ. Neglect any friction with the air. (a) Find the differential equation y" = f(y, y') satisfied by the function y, the displacement of the object from its equilibrium position, positive downwards. Write y for y(t) and yp for y' (t). y" : -9y Σ (b) Find r1, r2, roots of the characteristic polynomial of the equation above. r1,r2 = Зі, - 3і Σ (b) Find a set of real-valued fundamental solutions to the differential equation above. Y1(t) = cos(3t) Σ Y2(t) = sin(3t) Σ (c) At t = 0 the object is pulled down 1 cm and the released with an initial velocity downwards of 3/3 cm/sec. Find the amplitude A > 0 and the phase shift o E (-1, 7| of the subsequent movement. A = Σφ Σ
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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