What is the formula for max kinetic and max potential energy of a spring?

energy waves harmonic-oscillator spring

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Solution 1

I think you mean the energy of a particle attached to a spring. In that case the total energy is given by $$E=K+P=\frac{1}{2}mv^{2}+\frac{1}{2}kx^{2}$$ Where $m$ is the mass of the particle, $v$ is the velocity of the particle, and $x$ is the distance of the particle from the origin. So, the maximum of potential energy $P$ is given when the kinetic energy $K$ is zero (i.e. when $v=0$, in the turning points of the particle). In that case $E=P_{max}$. So $$E=P_{max}=\frac{1}{2}kx_{max}^{2}$$ here $x_{max}$ is the maximum stretch of the spring (i.e. the turning point, $v=0$).

In the other hand, the maximum of kinetic energy is given when $P=0$ (it occurs when $x=0$), i.e. $E=K_{max}$, or $$E=K_{max}=\frac{1}{2}mv_{max}^{2}$$ where $v_{max}$ is the maximun velocity of the particle, it occurs when the particle passes through the origin (i.e. when $x=0$).

Solution 2

Well, what's the equation for the kinetic and potential energies of the spring in general? Based on that, for what values of displacement will they be maximized?

56,250

Alpha

Updated on June 20, 2022

Comments

Alpha about 2 hours

What is the formula for max kinetic and max potential energy of a spring?
- dmckee --- ex-moderator kitten over 9 years
  
  Writing "concept" in the title doesn't really change this from a "give me the formula" question. As always you would use the definition of work and because your context is springs you would use Hooke's law for a first approximation.
Calc1DropOut about 8 years

Seems good but a better term for $x$ would be the displacement from the origin.
Ana S. H. about 8 years

That's exactly the meaning of my $x$.