To mathematically prove that the period and frequency of simple harmonic motion (SHM) are independent of amplitude, we can analyze the differential equation that describes SHM.
The equation of motion for SHM is typically given as:
d^2x/dt^2 = -(ω^2)x
Where: x is the displacement from the equilibrium position, t is time, ω is the angular frequency (ω = 2πf, where f is the frequency).
To find the period T and frequency f, we can look for solutions of the form:
x(t) = A cos(ωt + φ)
Where: A is the amplitude of the motion, and φ is the phase constant.
Differentiating x(t) twice with respect to time, we get:
d^2x/dt^2 = -Aω^2 cos(ωt + φ)
Substituting this back into the equation of motion, we have:
-Aω^2 cos(ωt + φ) = -(ω^2)A cos(ωt + φ)
Now, canceling out the common factors:
ω^2 = ω^2
This equation shows that the value of ω (angular frequency) remains the same regardless of the amplitude A. Therefore, the angular frequency ω is independent of the amplitude.
To find the period T and frequency f, we can use the relationship ω = 2πf. Since ω is constant, we can write:
2πf = ω
Solving for f, we have:
f = ω / (2π)
Since ω is constant, the frequency f is also independent of the amplitude.
Therefore, mathematically, we have shown that the period T and frequency f of simple harmonic motion are independent of the amplitude A.