To show that the total energy of a simple harmonic motion (SHM) is proportional to the square of the amplitude, we need to analyze the components of energy involved in SHM.
In SHM, the equation of motion for an object is given by:
x(t) = A * cos(ωt + φ)
Where:
- x(t) represents the displacement of the object from its equilibrium position at time t.
- A represents the amplitude of the motion.
- ω represents the angular frequency of the motion.
- φ represents the phase constant.
The total energy of the system consists of two components: kinetic energy (KE) and potential energy (PE).
- Kinetic Energy (KE): The kinetic energy of an object undergoing SHM is given by the formula:
KE = (1/2) * m * v^2
Where:
- m represents the mass of the object.
- v represents the velocity of the object.
To find the velocity, we differentiate the displacement equation with respect to time:
v(t) = dx/dt = -A * ω * sin(ωt + φ)
The maximum velocity (v_max) occurs when the sine function reaches its maximum value of 1:
v_max = A * ω
- Potential Energy (PE): The potential energy of an object undergoing SHM is given by the formula:
PE = (1/2) * k * x^2
Where:
- k represents the spring constant.
Substituting the displacement equation into the potential energy formula:
PE = (1/2) * k * (A * cos(ωt + φ))^2
Simplifying this expression:
PE = (1/2) * k * A^2 * cos^2(ωt + φ)
Using the trigonometric identity cos^2θ = (1/2) * (1 + cos(2θ)):
PE = (1/4) * k * A^2 * (1 + cos(2ωt + 2φ))
The average potential energy (PE_avg) over one complete cycle of motion is half of the maximum potential energy (PE_max):
PE_avg = (1/8) * k * A^2
Total Energy (E): The total energy (E) of the system is the sum of kinetic and potential energy:
E = KE + PE
E = (1/2) * m * v_max^2 + PE_avg
Substituting the expressions for v_max and PE_avg:
E = (1/2) * m * (A * ω)^2 + (1/8) * k * A^2
E = (1/2) * m * A^2 * ω^2 + (1/8) * k * A^2
E = A^2 * [(1/2) * m * ω^2 + (1/8) * k]
From this expression, we can see that the total energy (E) is proportional to the square of the amplitude (A^2). The coefficient [(1/2) * m * ω^2 + (1/8) * k] represents a constant value for a given system.
Therefore, the total energy of the SHM is indeed proportional to the square of the amplitude.