The evaluation of variation in oscillation time period of a simple pendulum as its mass varies proves a rich source of discussion in a physics class-room, overcoming erroneous notions carried forward by students as to what constitutes a pendulum's length due to picking up only the results of approximations and ignoring the rigorous definition. The discussion also presents a exercise for evaluating center of mass of geometrical shapes and system of bodies. In all, the pedagogical value of the problem is worth both theoretical and experimental efforts. This article discusses the theoretical considerations.
Deep Dive into The Moving Center of Mass of a Leaking Bob.
The evaluation of variation in oscillation time period of a simple pendulum as its mass varies proves a rich source of discussion in a physics class-room, overcoming erroneous notions carried forward by students as to what constitutes a pendulum’s length due to picking up only the results of approximations and ignoring the rigorous definition. The discussion also presents a exercise for evaluating center of mass of geometrical shapes and system of bodies. In all, the pedagogical value of the problem is worth both theoretical and experimental efforts. This article discusses the theoretical considerations.
What happens to a simple pendulum's oscillation time period with varying mass? This question is of pedagogical interest. An article in American Journal of Physics [1] addresses this issue from an experimental point of view, explaining variation in time period with oscillation of a burrette whose liquid content drips. From an introductory class point of view, the experiment misses many important issues by using a rigid pendulum instead of a simple pendulum. In this article, using theoretical considerations basic ideas of defining length of the pendulum (simple as it may look, the fine print is mostly overlooked), calculation of position of center of mass of a body and there-after a system of two bodies etc.
The oscillation time period of a simple pendulum is given as
where L is the length of the pendulum and g, the acceleration due to gravity. The absence of mass term in the above expression would either imply the incompleteness of the expression via approximations in derivation or some oversight. A brief review of the derivation shows
where the general expression of force is
giving
The linear velocity ‘v’, can be converted to angular velocity with a useful approximation (the small angle approximation), i.e. sinθ ≈ θ using the relation
It is from ω o = g L , that we obtain eqn (1). As per the equation, the undamped motion of the simple pendulum is indeed mass independent. But is the above derivation rigorous and exhaustive?
2 Time period of a Leaking Pendulum
The above derivation innocuously drops an important definition of force, defined as rate of change of momentum, i.e. F = dp dt = d(mv) dt . Eqn(2) follows only if mass is constant, which is not the case for a leaking pendulum. Thus, the derivation would require modifications.
Eqn( 5) is typically that of a damped pendulum and the time period would be given as
The expression typically shows how the time period would vary with variation in mass. However, from the observations of the burette experiment as also from our observations in case of a pendulum made with a hollow bob filled with water, the time period initially increases and then starts falling. Eqn(6) can not explain this observation considering the rate of change of mass of a leaking pendulum will always have values greater than or equal to zero.
To investigate further into the equation, we consider the length of the pendulum also to be changing with time. Thus, the above derivation changes from the point of eqn(4). That is,
giving an expression for time period as
For those who missed the rigorous definition of what constitutes the pendulum length would ponder how the length of the string used to suspend the bob would vary with time. That is, students carry a wrong notion that the length of the pendulum is the length of the string. This, however is only true if the bob is dense and considered a point mass with bob’s radius far smaller than the length of the string. Practically, this is not the case and the length of the pendulum would be length of the string and the radius of the bob. The radius is included since the whole mass of the bob is concentrated at its center, or its center of mass. The length of the pendulum is hence rigorously defined as distance between point of suspension to the center of mass of the pendulum.
In case of a leaking bob, the decreasing water would give a moving center of mass (see fig 1). The length of the pendulum then can be written as
x y o a a In the following section we derive an expression for the variable mass simple pendulum, eseenetially considering the bob to be a hollow shell filled with water and as it leaks the water level varies.
3 Where is the Center of Mass
The calculation of the center of mass (COM) of a body starts with a evaluation of the body’s mass. Since the shell in question has spherical symmetry, the integral to calculate the mass is best done in polar coordinates. Thus,
where ‘a o ’ and ‘a’ are the outer and inner radius of the shell whose density is ‘ρ shell ‘. The general expression for COM is given as
where M is the total mass of the body and m i is the mass of small volume (dx i dy i dz i ) of the body at r i from the origin. Applying this to the problem of shell (using eqn 9), we have
The COM is at the center of the shell.
The water mass filled in the shell, when the shell is filled can be considered to be a sphere of radius ‘a’, the shell’s inner radius. Now, consider disc of thickness ‘dz’ is cut from the sphere at a distance ‘z’ from the center. The disc has radius ‘r’ (see fig 2) and hence it’s area would be
The disc volume would be
The mass associated with this disc would be The net mass of the ‘water sphere’ hence would be
But radius of disc would depend on how far away from the center is the disc cut, hence r → r(z), which is obtained from simple rule r 2 = a 2 -z 2 . Hence,
In case the sphere is not completely filled then the calculations remain the same, however, the limits change. Say the water level is
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