I see now. ## \frac {dx_0} {dt} = 0 ## as for each differential time, there is no change in the rest position. This is because rest position is function of where the forces acting on the block are equal only. Right?nasu said:
Simple Harmonic Motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement from the equilibrium position and acts in the opposite direction. It is characterized by its sinusoidal nature, with displacement, velocity, and acceleration all varying sinusoidally with time. The motion can be described by the equation \( x(t) = A \cos(\omega t + \phi) \), where \( A \) is the amplitude, \( \omega \) is the angular frequency, and \( \phi \) is the phase constant.
The presence of an electric field exerts an additional force on the charged mass, which can alter the equilibrium position and the dynamics of the system. If the electric field is uniform, it adds a constant force to the system, shifting the equilibrium position but not affecting the nature of the SHM. If the electric field is non-uniform, it can introduce a position-dependent force, potentially complicating the motion and making it non-harmonic.
To derive the equation of motion, start by considering the forces acting on the charged mass. These include the restoring force of the spring, \( -kx \), and the electric force, \( qE \), where \( q \) is the charge and \( E \) is the electric field. Using Newton's second law, \( F = ma \), we get \( m\ddot{x} = -kx + qE \). Rearranging, we obtain the differential equation \( m\ddot{x} + kx = qE \). This is a standard form for SHM with an added constant force term.
For the system to exhibit SHM in the presence of an electric field, the electric field must be uniform so that it exerts a constant force on the charged mass. This ensures that the motion remains harmonic with a shifted equilibrium position. Additionally, the spring must obey Hooke's law, providing a linear restoring force proportional to the displacement. If these conditions are met, the system will exhibit SHM with a modified equilibrium position.
The frequency of the SHM is determined by the properties of the spring and the mass, and it remains unaffected by the presence of a uniform electric field. The angular frequency \( \omega \) is given by \(