Period of small oscillations

If the curve $E_\text{pot}(x)$ is a parabola, $E_\text{pot}(x) = a\,x^2 + b\,x + c$, the body oscillates between the two turning points with simple harmonic motion. This is a far-reaching result: a parabola-shaped potential well always produces the same type of motion, simple harmonic motion, regardless of the specific physical system that generated it. Whether it’s a spring, a pendulum for small angles, or any body near a potential-energy minimum, if the curve near the minimum is well approximated by a parabola the motion is harmonic and its period depends only on the “steepness” of the parabola and on the mass.

The oscillation period can be calculated directly from the coefficient of the parabola, without going through the force:

Principle — Oscillation period from $E_\text{pot}$

$\ev{T = 2\pi\sqrt{\frac{m}{2a\cos^2\alpha}}}$ where $a$ is the coefficient of $x^2$ in the parabola $E_\text{pot}(x)$, $m$ the mass of the body, and $\alpha$ the angle between the direction of the body’s motion and the direction along which we measure the coordinate $x$.

The coefficient $a$ measures how quickly the potential energy grows moving away from the minimum: a narrow, steep parabola (large $a$) corresponds to a strong restoring force and hence to fast oscillations, that is, a short period; a wide, flat parabola (small $a$) gives slow oscillations. Mass plays the opposite role: the heavier the body, the harder it is to reverse its motion, and the longer the period. The square root reflects the fact that quadrupling the mass, or halving the steepness, doubles the period.

The factor $\cos^2\alpha$ accounts for the fact that the coordinate $x$ used to measure the potential energy may not coincide with the direction along which the body actually moves.

Tip

If $x$ is along the motion, $\alpha = 0$ and $\cos^2\alpha = 1$. If $x$ is the horizontal projection of an inclined motion, $\alpha$ is the incline angle and the period increases.

Links

Topics: Lavoro ed energia Concepts: Moto armonico semplice · Periodo e frequenza · Energia potenziale elastica Skills: Approssimazione delle piccole oscillazioni Objects: Molla

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