The Measurement of Gravity

 

The Simple Pendulum with Friction

 

An expression for the period of a simple pendulum has already been developed using dimensional analysis and assumes the pendulum consists of a particle suspended from a rigid support by an inelastic string of negligible mass oscillating in vacuo with infinitely small amplitude. Consider the pendulum depicted in figure 3

 

Figure 3. The Simple Pendulum with Friction

Let () be the angular displacement. Now, the viscous retarding force will be proportional to the linear velocity of the bob and to the medium viscosity. The moment of this force about the origin O can be taken to be  where (k) is a proportionality constant and (l) is the string length. The weight will produce a moment {},that, if () is considered small will be equivalent to {}.

Hence the equation for the rotational motion about the axis through will be given by,

 

                          (1)

 

Dividing through by (ml²)

                                                       

 

                                             where  and

 

This is standard differential equation the solution of which is given by,

 

 

This equation represents ‘damped’ oscillations where the period is given by,

 

 

 

Figure 4. Graph of () against (t) for Decaying Oscillations

 

In the curve shown in figure 4, () is considered zero. On comparing the function for (T) with that for a simple pendulum  it is seen that,

 

 

In most cases (b) is small so expanding by the binomial theorem,

 

 

bearing in mind that   then  and thus,

 

 

Thus, for a pendulum swinging in air the period will slightly exceed that measured for a simple pendulum and thus, in determining (g) this should be taken into account.