The International Baccalaureate exploration in mathematics
Topic: Application of triple integrals in physics-calculation of period of a pendulum using
moment of inertia.
Personal code: gyz166
Session: May 2019
I declare that this work is my own work and is the final version. I have acknowledged each use of
the words or ideas of another person, whether written, oral or visual.
1
, 1. Introduction
My grandfather used to have a clock with pendulum and I was always baffled by the precision
of this instrument. What intrigued me was its design because altogether the pendulum all the time was
swinging at the same pace. When I was studying physics in high school I finally get to know how this
pace (now called period) was calculated. However, during physics classes various simplifications are
made in order to make problems more accessible mathematically. The pendulum was not an exception.
Presented as mathematical pendulum it was the ‘ideal’ object, it consisted of massless, inextensible
string and point mass (bob of the pendulum) attached at its end. These assumptions do not hold in
reality, unless the mass of the bob is significantly greater than that of the spring. However, the term
‘significantly’ always sounded vague and as I learned more physics it turned out that there is a way to
eliminate this inaccuracy. This new way included using moment of inertia, which fortunately was
taught as part of IB physics course. Nevertheless it was not perfect solution for me, the formulae were
just stated with little explanation, whereas in order to understand a concept, I think, it is worth to derive
certain formulae. In order to do so, with as little assumptions as possible, triple integrals need to be
used. That will allow for derivation of the formula for moment of inertia of physical (compound)
pendulum and then for using it to create formula for the period of such pendulum, what is the goal of
this exploration.
2. Difference between physical and mathematical pendulum
The mathematical pendulum is one consisting of a bob (being point mass) swinging on a
massless, inextensible rope, whereas in physical pendulum the rope or any other support isn’t massless
and as such requires additional physical description. The part of physics concerning itself with it, is
called the mechanics of rigid bodies. In this work the only useful concept will be moment of inertia,
however, it is worth to remember that there is much more to it than will be presented here.
The moment of inertia is the rotational equivalent of mass, that is, it is the measure of how
much the body will resist the force which causes the rotation. Its property which will be used later is
that it is equal to the distribution of mass around the axis of rotation (𝐼 = ∑ 𝑚𝑖 𝑟𝑖2 )(Tsokos, 2014).
For one particle motion it is easy but if the body is composed of many such particles then summing up
all the individual particles would be menial. Fortunately the summing up infinitely many objects does
sound familiar and that is because the basic principle of integration, as taught in high school, is the
same. A complicated shape (area under the curve) is ‘cut up’ into infinitely many pieces of infinitesimal
width and then their areas are added. However, physical objects are not one dimensional so simple
2
Topic: Application of triple integrals in physics-calculation of period of a pendulum using
moment of inertia.
Personal code: gyz166
Session: May 2019
I declare that this work is my own work and is the final version. I have acknowledged each use of
the words or ideas of another person, whether written, oral or visual.
1
, 1. Introduction
My grandfather used to have a clock with pendulum and I was always baffled by the precision
of this instrument. What intrigued me was its design because altogether the pendulum all the time was
swinging at the same pace. When I was studying physics in high school I finally get to know how this
pace (now called period) was calculated. However, during physics classes various simplifications are
made in order to make problems more accessible mathematically. The pendulum was not an exception.
Presented as mathematical pendulum it was the ‘ideal’ object, it consisted of massless, inextensible
string and point mass (bob of the pendulum) attached at its end. These assumptions do not hold in
reality, unless the mass of the bob is significantly greater than that of the spring. However, the term
‘significantly’ always sounded vague and as I learned more physics it turned out that there is a way to
eliminate this inaccuracy. This new way included using moment of inertia, which fortunately was
taught as part of IB physics course. Nevertheless it was not perfect solution for me, the formulae were
just stated with little explanation, whereas in order to understand a concept, I think, it is worth to derive
certain formulae. In order to do so, with as little assumptions as possible, triple integrals need to be
used. That will allow for derivation of the formula for moment of inertia of physical (compound)
pendulum and then for using it to create formula for the period of such pendulum, what is the goal of
this exploration.
2. Difference between physical and mathematical pendulum
The mathematical pendulum is one consisting of a bob (being point mass) swinging on a
massless, inextensible rope, whereas in physical pendulum the rope or any other support isn’t massless
and as such requires additional physical description. The part of physics concerning itself with it, is
called the mechanics of rigid bodies. In this work the only useful concept will be moment of inertia,
however, it is worth to remember that there is much more to it than will be presented here.
The moment of inertia is the rotational equivalent of mass, that is, it is the measure of how
much the body will resist the force which causes the rotation. Its property which will be used later is
that it is equal to the distribution of mass around the axis of rotation (𝐼 = ∑ 𝑚𝑖 𝑟𝑖2 )(Tsokos, 2014).
For one particle motion it is easy but if the body is composed of many such particles then summing up
all the individual particles would be menial. Fortunately the summing up infinitely many objects does
sound familiar and that is because the basic principle of integration, as taught in high school, is the
same. A complicated shape (area under the curve) is ‘cut up’ into infinitely many pieces of infinitesimal
width and then their areas are added. However, physical objects are not one dimensional so simple
2
