A one-dimensional system of two bodies of mass m₁ and m2 and three springs with Hooke's law constants k₁, k2, and k3 is fixed between two walls as shown in the figure. The displacements of the two bodies from their equilibrium positions are î₁ and £2, as shown in the figure. m 1 Foooooooooooo k₂ X₁ M k₁ a) Write down the total force acting on each of the two bodies. (Note it is not necessary to determine the equilibrium positions relative to the walls.) b) Write down the equations of motion for the two bodies. c) Hereafter assume m₁ = m, m₂ = 2m and k₁= k2 = k3 = k. Show the equations of motion in the matrix form is where M and G are 2 × 2 matrices with - Mx + Gx = 0 m (TO 2) 0 X2 and G = m2 2k - k -k 2k k3
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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