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THE SIMPLE PENDULUM |
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The simple pendulum provides an interesting example of harmonic motion, as it exhibits simple harmonic motion only for small angles of oscillation. In this section, we will take a look at harmonic motion of the simple pendulum for arbitrary angles of oscillation. |
The simple pendulum is indeed a very simple mechanical system, and it is illustrated in the following figure:
In essence, it consists of a mass attached to a piece of string, which is in turn fixed so that the mass can swing in a vertical plane. Now, it is not difficult to show that the equation of motion for the mass is
There are various ways to arrive at this result. One is to use the rotational equivalent of Newton’s Second law, namely that torque equals moment of inertia multiplied by angular acceleration. Taking moments about the axis of rotation, the vertical gravitational force on the mass M gives rise to a torque
The moment of inertia of the mass M about the axis of rotation is just
Application of the rotational form of Newton’s Second Law about the axis of rotation then gives the equation of motion.
We can begin by noting that if the angle of oscillation is small, then the following holds:
This is the characteristic equation of simple harmonic motion. Thus, when the angle of oscillation θ is small, the simple pendulum exhibits simple harmonic motion.
To solve the full equation for arbitrary angles of oscillation, we begin by multiplying both sides of the equation by the first derivative of the angle θ with time, and note that the equation can then be rewritten in the form
where we have defined
This equation can be integrated to give
where A is a constant. The constant A can be determined by assuming that the angular velocity is zero at the initial angle of release, which we will call θ0. This gives
and hence we can write
The solution of this equation then takes the following form:
The time period T of the oscillations is given by
This expression arises because the time taken for the motion from θ0 to 0 requires, by symmetry, 1/4 of the total time period.
Unfortunately, the above integrals can not be evaluated in terms of elementary functions such as sine and cosine. However, making the following variable changes
and noting that the following relationships then hold
The solution can be written in the form
and the time period T of the oscillations is given by
Both of these integrals are known as “incomplete elliptic integrals of the first kind”, and the values can be obtained either from tabulations or by numerical integration.
It is of interest to compare the time period of oscillations for arbitrary angles of oscillation with the time period for simple harmonic oscillations, in which the angle of oscillation is small. The time period for simple harmonic oscillations is given by
and hence the ratio of the time period for arbitrary oscillations to this time period is
Note that if k is small, then the integrand is very nearly unity, and so the ratio of time periods is about unity, as would be expected. Some values of the ratio for various initial starting angles θ0 are listed below:
θ0 = 10 degrees ratio = 1.002
θ0 = 90 degrees ratio = 1.18
θ0 = 135 degrees ratio = 1.53
Thus for an initial starting angle of 135 degrees, the time period is 53% greater than the time period for simple harmonic motion at small angles of oscillation.