Physics Situation: Unusual Start Position for a Ideal Pendulum

A ideal pendulum has been released from the horizontal position,

4 min readDec 12, 2022

after how long will it reach position C?

1️⃣Analysis

According to the drawing there is a circular trajectory, so we are facing a rotational movement. The indication of acting forces requires the use of the laws of classical mechanics, that is, Newton’s Laws. And since there is a constraint acting on the system, this leads to the practical use of D’Alemberg’s principle, that is, a static analysis is used, considering the same forces acting, now, under a condition of equilibrium. In this way we have:

where the first term corresponds to a parallel trigonometric component of inertial nature and the second term corresponds to an active force calculated based on the centripetal acceleration. Thus, it remains to describe the movement perpendicular to that parallel component, in such a way that the relation is valid:

where, in that expression, its right-hand term is representing the effect of a tangential force expressed in its second term according to Newton’s second law.

2️⃣Solution

Mathematically, the variables of the problem will be involved by a system of equations:

where m is the literal value of the mass attached to the pendulum string, g is the acceleration due to gravity and and R is the length of the pendulum string.

Let’s work on the second equation of the system.

and, avoiding rhetorical steps, we arrive at the expression:

and already aiming to integrate this into the third equation, calculate the mathematical derivative of v as a function of θ

and

Now, working on the third equation, we have

and

Therefore,

Unfortunately the secant function is not defined for two trigonometric quadrants, i.e.

one of which corresponds to one of the limits of integration of our expression above, which is the third quadrant. In this case we will use a physics strategy to solve the necessary integral and thus obtain the result.

We divide the third quadrant into two segments, one corresponding to 10 trigonometric degrees and the other corresponding to 80 trigonometric degrees. In such a way that there will now be two intervals of integration, whose summed results will be the answer to the problem, that is,

For an angle of 10 degrees, our pendulum approaches the limit of the simple pendulum, whose theory says that the relation is valid as period of oscillation:

and so, we will have

and in the most simplified form:

The remaining term in square brackets will be calculated numerically. We will use the wxMaxima® platform, and its quad_qags function. The function is invoked using the following syntax:

→ numer: true$
%pi - quad_qags (sqrt(sec(θ)), θ, 0,-4*%pi/9)[1];

whose execution results

(%i2) numer: true$
%pi - quad_qags (sqrt(sec(θ)), θ, 0,-4*%pi/9)[1];
(%o2) 4.927682708875202