CBSE Class 11-science Answered

when an object is subjected to a restoring force F that is proportional to displacement,

then the object is undergoing simple harmonic motion.

Let m be the mass of the object .

If restoring force F  is proportional to displacement x , then it is mathemetically written as

F ∝ x

F = - k x  ( negative sign is used because force is restoring force )

where k is proportionality constant and it is known as force constant

force F = mass × acceleration

Above expression can be written as

  .........................(1)

where ω² = k/m .

For above second order differential equation , the solution is written as

x = a cos(ωt) + b sin(ωt)  ......................(2)

where a and b are constants ,

If we choose the constants so that

a/b = tan φ ;  a =  A cos φ  ; b = A sin φ

then eqn.(2) will be written as

x = A [ cosφ  cos(ωt)+ sinφ sin(ωt)]

x = A cos( ωt - φ )

where A is amplitude , ω is angular frequency and φ is initial phase at t = 0