To demonstrate that the frequency of a simple harmonic motion (SHM) is independent of the amplitude, we can use the equation for the period of an oscillating system. The period is the reciprocal of the frequency and is the time taken for one complete oscillation.
For a mass-spring system undergoing SHM, the period (T) can be expressed as:
T = 2π * √(m/k),
where: T is the period, π is a mathematical constant (approximately 3.14159), m is the mass of the object undergoing SHM, and k is the spring constant.
From this equation, we can see that the period (and hence the frequency) of the oscillation is only dependent on the mass and the spring constant, and it is independent of the amplitude of the motion.
To illustrate this, let's consider two mass-spring systems with different amplitudes but the same mass and spring constant. The periods of oscillation for both systems will be the same, implying that the frequencies are also identical.
For example, let's assume we have two mass-spring systems:
System 1: Amplitude = A1 Mass = m Spring constant = k
System 2: Amplitude = A2 (where A2 ≠ A1) Mass = m Spring constant = k
Both systems will have the same period (T) according to the formula mentioned above:
T = 2π * √(m/k)
Since the masses and spring constants are the same, the periods for both systems will be equal. Thus, the frequencies (f = 1/T) will also be the same, regardless of the amplitude. This demonstrates that the frequency of simple harmonic motion is independent of the amplitude.