Solutions to Simple Harmonic Motion second order differential equation

  • #1
Trollfaz
140
14
All simple harmonic motion must satisfy
$$\frac{d^2s}{dt^2}=-k^2s$$
for a positive value k.
The most well known solution is the sinusoidal one
$$ s=Acos/sin(\omega t + \delta)$$
A is amplitude, ##\omega##is related to frequency and ##\delta## is phase displacement.
My lecturer said that there might be other functions that satisfy the second order differential equation and I would like to know some other solution to the equation
 
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  • #2
A second order differential equation has two independent solutions. With the amplitude and phase parameters you are covering both of those.
 
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FAQ: Solutions to Simple Harmonic Motion second order differential equation

What is the general form of the second-order differential equation for simple harmonic motion?

The general form of the second-order differential equation for simple harmonic motion is \( \frac{d^2x}{dt^2} + \omega^2 x = 0 \), where \( x \) is the displacement, \( t \) is time, and \( \omega \) (omega) is the angular frequency of the motion.

How do you find the general solution to the simple harmonic motion differential equation?

The general solution to the simple harmonic motion differential equation \( \frac{d^2x}{dt^2} + \omega^2 x = 0 \) is \( x(t) = A \cos(\omega t) + B \sin(\omega t) \), where \( A \) and \( B \) are constants determined by initial conditions.

What are the initial conditions and how do they affect the solution?

The initial conditions typically specify the initial displacement \( x(0) \) and initial velocity \( \frac{dx}{dt}(0) \). These conditions are used to solve for the constants \( A \) and \( B \) in the general solution. For instance, if \( x(0) = x_0 \) and \( \frac{dx}{dt}(0) = v_0 \), then \( A = x_0 \) and \( B = \frac{v_0}{\omega} \).

What is the physical interpretation of the constants \( A \) and \( B \) in the solution?

The constant \( A \) represents the amplitude of the cosine component of the motion, while \( B \) represents the amplitude of the sine component. Together, they determine the overall amplitude and phase of the oscillation. The amplitude \( \sqrt{A^2 + B^2} \) represents the maximum displacement from the equilibrium position.

How does damping affect the solution to the simple harmonic motion differential equation?

Damping introduces a term proportional to the velocity into the differential equation, resulting in \( \frac{d^2x}{dt^2} + 2\beta \frac{dx}{dt} + \omega^2 x = 0 \). The solution to this damped harmonic motion depends on the damping coefficient \( \beta \). For underdamped motion (\( \beta < \omega \)), the solution is \( x(t) = e^{-\beta t} (A \cos(\omega' t) + B \sin(\omega' t)) \), where \( \omega' = \sqrt{\omega^2 - \beta^2} \). For critically damped (\( \beta = \omega \)) and overdamped

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