Equation of a simple harmonic motion(SHM)

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SUMMARY

The equation of simple harmonic motion (SHM) is defined as y = a sin(ωt) or x = a cos(ωt), derived from the differential equation m(d²x/dt²) = -kx. This relationship indicates that the restoring force is proportional to displacement, leading to the conclusion that the solutions to this equation are sine and cosine functions. The angular frequency ω₀ is expressed as ω₀ = √(k/m), where k is the spring constant and m is the mass of the oscillator. Understanding these derivations enhances comprehension of SHM beyond rote memorization.

PREREQUISITES
  • Understanding of Newton's second law of motion
  • Familiarity with differential equations
  • Knowledge of trigonometric functions, specifically sine and cosine
  • Basic concepts of oscillatory motion and restoring forces
NEXT STEPS
  • Study the derivation of the differential equation for simple harmonic motion
  • Explore the properties of sine and cosine functions in relation to SHM
  • Learn about the energy conservation in simple harmonic oscillators
  • Investigate the applications of SHM in real-world systems, such as pendulums and springs
USEFUL FOR

Students of physics, educators teaching mechanics, and anyone interested in the mathematical foundations of oscillatory motion will benefit from this discussion.

johncena
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According to my textbook, the equation of a simple harmonic motion(SHM) is given by,
y = a sin\omegat
or, x = a cos\omegat
But how did these equations came?
Like this there are somany equations in my textbook without derivations.And studying these byheart makes the subject very boring.
 
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A simple harmonic oscillator feels a restoring force that is proportional to it's displacement, so Newton's second law tells us:

<br /> <br /> m\frac{d^2x}{dt^2} = -kx<br /> <br />

or

<br /> <br /> \frac{d^2x}{dt^2} = -\omega_o^2x<br /> <br />

where

<br /> \omega_o = \sqrt{\frac{k}{m}}<br />

Now, there are some techniques for solving this differential equation, but we can just think about it intuitively. What function, if you take it's derivative twice, gives back the negative of the same function times a constant. Well cosine and sine both posses those properties. So those are both solutions, and any linear combination of them will be a solution.
 

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