Example of oscillations | Learn Physics

Example of oscillations

Example: A thin rod of length L and area of cross-section S is pivoted at its lowest point P inside a stationary, homogeneous and non viscous liquid (fig.1). The rod is free to rotate in a vertical plane about a horizontal axis passing through P. The density d, of the material of the rod is smaller than the density d₂ of the liquid. The rod is displaced by a small angle θ from its equilibrium position and then released. Show that the motion of the rod is simple harmonic and determine its angular frequency in terms of the given parameters.

 

Solution:

When the rod is displaced slightly through an angle θ with the vertical Fig.2, the effective upward force acting at the middle point G of the rod is:

                               F= upward thrust – weight of rod

                               = weight of liquid displaced – weight of rod

                                  =LSd₂g – LSd₁g =LSg(d₂-d₁)

Moment of the couple (i.e. torque) tending to bring the rod to its original position is:

 

                            Torque = F × KG

                         =  LSg(d₂-d₁) ×L sinθ/2

                        = [Sg(d₂-d₁)L²/2] θ …………………………(i)

Θ is small

 

So torque is proportional to θ. As this torque is restoring one, hence if the rod is left free it will execute SHM. Compare (I)

     Torque = k θ, we have

Spring factor = Sg(d₂-d₁)L²/2

 

Here, inertia factor = moment of inertia of the rod about axis at P

                           = ML²/2+M(L/2)² = ML²/3 =L³Sd₁/3

                As angular frequency in S.H.M. is

                     ω= √(spring factor/inertia factor)=√[3g(d₂-d₁)/2Ld₁]