Problem — Reading the E(x) graph of a pendulum

Graph reading: $E(x)$ for a pendulum. The graph shows $E_k$ and $E_p$ as functions of the position $x$ of a pendulum: $E_p$ is a parabola with a minimum at $x = 0$, $E_k$ is its reflection. Identify $E_\text{tot}$ on the graph and discuss conservation.

Solution

Identifying $E_\text{tot}$. The total mechanical energy is the point-by-point sum of the two curves:

$E_\text{tot}(x) = E_k(x) + E_p(x)$

In the graph $E_p$ is the parabola $E_p \propto x^2$ (minimum at the centre) and $E_k$ is its reflection. Adding them, the $x^2$ term cancels: the result is a horizontal line (dashed, above both curves). This constant line is $E_\text{tot}$.

Point-by-point reading:

• At the centre ($x = 0$): $E_p$ is minimum (zero), so $E_k$ is maximum. The pendulum passes through the lowest point at maximum speed.
• At the extremes (turning points): $E_k = 0$ (the pendulum is momentarily at rest), so $E_p$ is maximum. Here the motion reverses.
• At every intermediate point: whatever one curve loses, the other gains, and the sum stays the same.

Conservation:

$\ev{E_\text{tot} = E_k(x) + E_p(x) = \text{constant} \ \Rightarrow\ \text{horizontal line}}$

Physical reading. The graph visualises conservation of mechanical energy: energy is neither created nor destroyed, it is continuously transferred between kinetic and potential as the pendulum swings. The horizontal line of $E_\text{tot}$ is the graphical signature of this conservation (valid for the ideal pendulum, with no friction).

Collegamenti

Argomenti: Work and energy Concetti: Conservation of mechanical energy Competenze: Reading graphs Oggetti: Simple pendulum