Comprehensive Study Notes on Torsion Pendulum Dynamics

Analytical Study of the Torsion Pendulum Motion

The primary objective of this study is to perform a kinetic analysis of a torsion pendulum (نواس الفتل) to determine the specific nature of its movement and to derive the mathematical law for its natural period ( $T_0$ ). The torsion pendulum typically consists of a rigid rod suspended by a metallic wire from its center. When the rod is rotated through a certain angle and released, the torsion of the wire provides a restoring torque that causes the rod to oscillate.

Classification of Influencing Forces and Torques

To begin the dynamical study, we first identify all external forces acting upon the suspended rod. These forces are critical for establishing the equilibrium and motion equations within the laboratory frame of reference. The forces identified are as follows:

The weight of the rod, denoted as $\mathbf{W}$ . This force acts vertically downwards from the center of gravity of the rod.
The tension of the suspension wire, denoted as $\mathbf{T}$ . This force acts vertically upwards along the axis of the wire.
The torsion couple (مزدوجة الفتل), which produces a restoring torque denoted by $\Gamma$ . This torque is a result of the elastic properties of the wire when twisted.

The Fundamental Law of Rotational Dynamics

We apply the basic principle of rotational dynamics, also known as Newton's second law for rotation or the angular acceleration theorem. The law states that the sum of the moments (torques) of all external forces acting on a body about a fixed axis of rotation ( $\Delta$ ) is equal to the product of the body's moment of inertia ( $I_{\Delta}$ ) and its angular acceleration ( $\alpha$ ). The mathematical expression is:

$\sum \Gamma_{/ \Delta} = I_{\Delta} \alpha$

Breaking down the sum of torques for our specific case:

$\Gamma_{W / \Delta} + \Gamma_{T / \Delta} + \Gamma_{\text{torsion}} = I_{\Delta} \alpha$

In this system, the lines of action for both the weight ( $\mathbf{W}$ ) and the tension ( $\mathbf{T}$ ) of the wire coincide with the axis of rotation ( $\Delta$ ). By definition, if a force's line of action passes through the axis of rotation, its torque is zero. Therefore:

$\Gamma_{W / \Delta} = 0$

$\Gamma_{T / \Delta} = 0$

Substituting these values into the dynamic equation, we are left with the torque provided by the torsion couple:

$0 + 0 - K \theta = I_{\Delta} \alpha$

Here, $-K \theta$ represents the restoring torque of the torsion wire, where $K$ is the torsion constant of the wire and $\theta$ is the angular displacement.

Formal Derivation of the Differential Equation

By rearranging the dynamic equation, we can isolate the angular acceleration ( $\alpha$ ), which is the second derivative of the angular displacement with respect to time ( $\theta''$ or $\frac{d^2\theta}{dt^2}$ ). The equation becomes:

$-K \theta = I_{\Delta} \theta''$

This leads to the standard form of the differential equation for the torsion pendulum:

$\theta'' = -\frac{K}{I_{\Delta}} \theta$

This is a second-order linear homogeneous differential equation. Because the acceleration is proportional to the negative of the displacement, it characterizes a specific type of periodic motion.

Analytical Solution and the Nature of Motion

A general solution for this type of differential equation is a sinusoidal function (جيبية) of time. The proposed solution for the angular position is:

$\theta(t) = \theta_{\max} \cos(\omega_0 t + \phi)$

In this expression, $\theta_{\max}$ represents the maximum angular amplitude, $\omega_0$ is the natural angular frequency of the oscillation, and $\phi$ is the phase constant at the initial time ( $t = 0$ ).

Verification of the Solution via Time Derivatives

To confirm that the proposed sinusoidal solution is valid, we must differentiate the displacement function twice with respect to time and substitute it back into the differential equation.

First derivative (Angular Velocity): $\theta'(t) = -\omega_0 \theta_{\max} \sin(\omega_0 t + \phi)$

Second derivative (Angular Acceleration): $\theta''(t) = -\omega_0^2 \theta_{\max} \cos(\omega_0 t + \phi)$

Since $\theta(t) = \theta_{\max} \cos(\omega_0 t + \phi)$ , we can simplify the second derivative to:

$\theta''(t) = -\omega_0^2 \theta$

By comparing this result to the original differential equation derived from rotational dynamics ( $\theta'' = -\frac{K}{I_{\Delta}} \theta$ ), we find that the solution is valid provided that the following condition is met:

$\omega_0^2 = \frac{K}{I_{\Delta}}$

Since both the torsion constant ( $K$ ) and the moment of inertia ( $I_{\Delta}$ ) are positive and constant physical values, the term $\frac{K}{I_{\Delta}}$ is always greater than zero (> 0). This confirms that the motion of the torsion pendulum is Simple Harmonic Rotational Motion (حركة جيبية دورانية).

Scientific Curiosity: The Mpemba Effect

As a concluding note of interest, the transcript mentions an observation regarding thermodynamics: hot water freezes faster than cold water. This phenomenon is known in physics as the Mpemba effect. While it may seem counterintuitive, various factors such as evaporation, convection, and dissolved gases contribute to this observation in specific conditions.