Question

The differential equation of SHM for a seconds pendulum is

Answer 1

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Answer 2

Answer:

The differential equation of SHM for a seconds pendulum is                          $\frac{d^{2}x }{dt^{2} }$ + (k/m)x = 0.    

Explanation:

Simple Harmonic motion:

• Simple harmonic motion, or SHM, is an oscillatory motion in which the acceleration of the particle at any position is directly proportional to the displacement from the mean position.
• This particular oscillatory motion is a special case. While all oscillatory motions are oscillatory and periodic, not all oscillatory motions are Simple Harmonic Motions (SHM).
• The linear periodic motion of a body where the restoring force is directly proportionate to the displacement from the equilibrium position and is always oriented towards the equilibrium position or mean position.
• Additionally, while all periodic motions are not simple harmonic motions, all simple harmonic motions are periodic in nature.
• Have you ever seen a pendulum, for example? It swings back and forth in the same direction. They are oscillations, these movements. Pendulum oscillations are an illustration of straightforward harmonic motion.  

• Let F represent the force and x represent the seconds pendulum displacement from equilibrium.

                                  F ∝ -x

                                  F = -kx

• Therefore, F= - kx will be the restoring force (the negative sign indicates that the force is in the opposite direction).
• The force constant, or k, is present in this situation.
• In the SI and CGS systems, it is measured in N/m and dynes/cm, respectively.

                             F + kx = 0

                           ma + kx = 0

                  since a = $\frac{d^{2}x }{dt^{2} }$

                        m$\frac{d^{2}x }{dt^{2} }$ + kx = 0

                          divide the above equation with 'm'.

                        $\frac{d^{2}x }{dt^{2} }$ + (k/m)x = 0

           This the differential equation of SHM for a seconds pendulum.

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