To determine the time period of oscillation for the given system, we need to analyze the forces and torques acting on the disc when it is slightly displaced. The setup involves a disc with a point charge at its center and another charge positioned vertically above it. When the disc is displaced horizontally, it experiences a restoring force due to the electric interaction between the charges, which leads to oscillatory motion. Let's break this down step by step.
Understanding the Forces at Play
When the disc is displaced, the point charge -Q at the center of the disc exerts an electric force on the charge +Q located at a distance 'l' above it. This force can be calculated using Coulomb's law:
- The electric force \( F \) between two point charges is given by:
- \( F = \frac{k \cdot |Q_1 \cdot Q_2|}{r^2} \), where \( k \) is Coulomb's constant, \( Q_1 \) and \( Q_2 \) are the magnitudes of the charges, and \( r \) is the distance between them.
In our case, the distance between the charges when the disc is displaced horizontally by a small distance \( x \) becomes \( \sqrt{x^2 + l^2} \). Thus, the force acting on the charge +Q can be expressed as:
\( F = \frac{k \cdot Q^2}{x^2 + l^2} \)
Torque and Angular Motion
When the disc rolls without slipping, the displacement creates a torque about the center of mass. The torque \( \tau \) due to the electric force can be calculated as:
\( \tau = F \cdot R \)
Substituting the expression for \( F \), we have:
\( \tau = \frac{k \cdot Q^2}{x^2 + l^2} \cdot R \)
Relating Torque to Angular Acceleration
The angular acceleration \( \alpha \) of the disc is related to the torque by the equation:
\( \tau = I \cdot \alpha \)
Here, \( I \) is the moment of inertia of the disc, which is given by:
\( I = \frac{1}{2} M R^2 \)
Substituting for \( \tau \) and rearranging gives us:
\( \frac{k \cdot Q^2}{x^2 + l^2} \cdot R = \frac{1}{2} M R^2 \cdot \alpha \)
Finding the Time Period of Oscillation
For small displacements, we can use the small angle approximation, where \( \alpha \) can be expressed in terms of linear displacement \( x \) as:
\( \alpha = \frac{d^2x}{dt^2} \)
Substituting this into our torque equation, we can derive the equation of motion. The resulting differential equation will resemble that of simple harmonic motion:
\( \frac{d^2x}{dt^2} + \omega^2 x = 0 \)
Where \( \omega^2 \) is the angular frequency. The time period \( T \) of oscillation is given by:
\( T = 2\pi \sqrt{\frac{I}{\tau}} \)
After substituting the expressions for \( I \) and \( \tau \) and simplifying, we arrive at the final expression for the time period:
\( T = 2\pi \sqrt{\frac{6 \pi \epsilon M l^3}{Q}} \)
Final Thoughts
This result shows how the interplay between electric forces and rotational dynamics leads to oscillatory motion in the system. The time period depends on the mass of the disc, the distance from the charge, and the magnitude of the charges involved. Understanding these relationships is crucial in fields like electrostatics and dynamics.
