Linear Simple Harmonic Motion.

Simple Harmonic Motion :- 

☛ Linear Simple Harmonic Motion:-

                The motion in which object moves 'to and fro' along the straight line is called linear simple harmonic motion.

E.g:-

• Oscillation of pendulum of clock.
• Oscillation of needle of sewing Machine.
• Oscillation of spring loaded with mass.

This all motions are harmonic in Nature

★ Harmonic functions are denoted in terms of sin𝛉  & cos𝛉. therefore whatever equations we will derive in this topic are only in terms of sin 𝛉 & cos 𝛉.
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Description:-

         Spring mass oscillator:-

           Consider a body of mass m attached to one end of an ideal spring of force constant k and free to move over a frictionless horizontal surface as shown in above fig.

           🤩 The 1st diagram shows that the mass attached to a spring is at the mean position :-            

1)  At the mean position P.E=0 & It is a constant or stable position of mass.

2)  It is also in stable equilibrium so it's displacement is 0.

3)  Now from here if we pull mass the spring attached to it expands. 

4)  Then at extreme position direction of displacement is towards right side & restoring force (Fr) is towards mean position due to its tension in the string.

            🤩  The 2nd diagram shows that the mass attached to a spring is at the right extreme position :-

 1)  At the right extreme position the force of the string(Fr) is towards left which opposite to displacement because our displacement is towards right side. 

2)   When we remove the mass that we displaced some time before compresses due to Moment of Inertia (MI).

            🤩  The 3rd diagram shows that the mass attached to a spring is at the left extreme position :-

1)  But at this time the after the removal of mass from right extreme position the displacement is towards left extreme & direction of displacement (x) and restoring force (Fr) is towards mean position.

                          

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                Greater the restoring force greater the displacement & lower the restoring force lower the displacement is.

                             ∴ Restoring force (Fr) ∝ displacement(x)

      

                At every case, the force, is  called a restoring force . If (x) is displacement , the restoring force is given by 

                              F = -kx

                where,

                         k= force constant or spring constant. k depends upon on thickness of spring or quality of spring. so, S.I unit of force constant is N/m and C.G.S unit is dyne/cm.

                         x=displacement