A particle is oscillating in simple harmonic motion. The time required for the particle to travel through one complete cycle is equal to the period of the motion, no matter what the amplitude is. But how can this be, since larger amplitudes mean that the particle travels farther?

oh and could you please explain

Maximum speed during oscillation is proportional to amplitude. That keeps the period independent of amplitude

Although larger amplitudes in simple harmonic motion do result in the particle traveling a greater distance, the time required for the particle to complete one full cycle remains constant. This is because the period of simple harmonic motion is determined by the nature of the restoring force acting on the particle, not by the amplitude.

In simple harmonic motion, the force acting on the particle is directly proportional to the displacement from the equilibrium position, and the direction of the force is always opposite to the displacement. This results in the particle experiencing an acceleration towards the equilibrium position. As the particle moves away from the equilibrium position, the force acting on it increases, effectively accelerating it towards the equilibrium position. As the particle approaches the equilibrium position, the force decreases, decelerating the particle.

Since the acceleration of the particle is determined solely by the restoring force, and not by the amplitude, the period of the motion remains constant. The period is defined as the time taken for the particle to complete one full cycle of oscillation, regardless of the amplitude. Therefore, even though the particle may travel a greater distance for larger amplitudes, it still completes one full cycle in the same amount of time as for smaller amplitudes.

The time required for a particle to travel through one complete cycle in simple harmonic motion is indeed equal to the period of the motion, regardless of the amplitude. This might seem counterintuitive, as a larger amplitude suggests that the particle travels a greater distance. However, it's important to distinguish between the time required to complete one cycle and the distance traveled during that cycle.

To understand why the period remains constant, let's break down the motion of a particle in simple harmonic motion. In this type of motion, the particle moves back and forth about an equilibrium position with a restoring force proportional to its displacement. The resulting motion is a sinusoidal curve.

The period of the motion, denoted by T, refers to the time it takes for the particle to complete one full cycle. It is primarily dependent on the restoring force and the mass of the particle. Mathematically, the period is given by T = 2π√(m/k), where m represents the mass of the particle and k is the force constant (or spring constant) of the system.

The amplitude, denoted by A, refers to the maximum displacement of the particle from its equilibrium position. It tells us how far the particle can move from its central position. The relationship between the amplitude and the distance traveled during one cycle is given by the equation D = 2A, where D represents the total distance traveled.

Now, the reason the period remains constant is because the velocity of the particle changes as it moves through the cycle. When the particle is at its maximum displacement (amplitude), its velocity is momentarily zero. As it approaches the equilibrium position, its velocity increases, reaching a maximum when it passes through the central position. As it moves further away from the equilibrium, its velocity decreases until it again becomes zero at the maximum displacement in the opposite direction.

This variation in velocity compensates for the larger distance traveled at larger amplitudes. Even though the particle covers a greater distance from its equilibrium position, the time required to move through each segment of the cycle is adjusted accordingly. The particle travels faster at larger amplitudes but spends less time in each portion of the cycle.

To summarize, the period of a particle in simple harmonic motion remains constant because the variation in velocity compensates for the larger distance traveled at larger amplitudes. This balance ensures that the particle completes one full cycle in the same amount of time, regardless of the amplitude.