First slide

Stable, Unstable and Neutral equilibrium

Question

The force F acting on a body moving along x-axis varies with the position x of the particle as shown in the graph. The body is in stable equilibrium at

Difficult

A

$x = x_{1}$
B

$x = x_{2}$
C

$both x_{1} and x_{2}$
D

$neither x_{1} nor x_{2}$

Solution

$\begin{array}{l} If \frac{d^{2} U}{d x^{2}} > 0, then it is the position of stable equillibrium \\ If \frac{d^{2} U}{d x^{2}} < 0, then it is the position of unstable equillibrium \\ {At x}_{1}, \frac{d F}{d x} > 0 \Rightarrow \frac{d}{d x} (- \frac{d U}{d x}) > 0, i.e., \frac{d^{2} U}{d x^{2}} < 0 \\ x_{1} is the position of unstable equillibrium \\ {At x}_{2}, \frac{d F}{d x} < 0 \Rightarrow \frac{d}{d x} (- \frac{d U}{d x}) < 0, i.e., \frac{d^{2} U}{d x^{2}} > 0 \\ x_{2} is the position of stable equillibrium \end{array}$

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