Solutions to Simple Harmonic Motion second order differential equation

  • #1
Trollfaz
137
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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1. What is the general solution to the Simple Harmonic Motion second order differential equation?

The general solution to the Simple Harmonic Motion second order differential equation is given by x(t) = A*cos(ωt) + B*sin(ωt), where A and B are constants determined by the initial conditions of the system, and ω is the angular frequency of the motion.

2. How do you determine the angular frequency in the Simple Harmonic Motion equation?

The angular frequency ω in the Simple Harmonic Motion equation is determined by the equation ω = √(k/m), where k is the spring constant and m is the mass of the object undergoing harmonic motion.

3. What are the units of the spring constant in the Simple Harmonic Motion equation?

The units of the spring constant k in the Simple Harmonic Motion equation are in Newtons per meter (N/m), as it represents the force required to stretch or compress the spring by a unit distance.

4. How does the amplitude of the motion affect the Simple Harmonic Motion equation?

The amplitude of the motion affects the Simple Harmonic Motion equation by determining the maximum distance the object can move from its equilibrium position. It is represented by the coefficient A in the general solution x(t) = A*cos(ωt) + B*sin(ωt).

5. Can the Simple Harmonic Motion equation be used to model real-world systems?

Yes, the Simple Harmonic Motion equation can be used to model a wide range of real-world systems, such as pendulums, vibrating strings, and mass-spring systems. It provides a simple and accurate description of periodic motion in these systems.

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