# GRE Physics GR 1777 Problem Solution

## 046. Classical Mechanics (Hooke\’s law)

$latex \\vec{F} = – k \\Delta x \\ \\ \\text{where } k \\text{ is a force constant and } x \\text{ is a displacement (change in length of a spring).}$

$latex U = \\frac{1}{2} k x^2 = \\frac{1}{2} (kx) x = \\frac{1}{2} |\\vec{F}| x$

### Solution

• Extension

The same external force is applied to both springs,

$latex |\\vec{F}| = k\\Delta x = k_1 \\Delta x_1 = k_2 \\Delta x_2$

$latex \\text{Since } k_1 > k_2, \\text{ then } \\Delta x_1 < \\Delta x_2$

• Stored Potential Energy

$latex U_1 = \\frac{1}{2} k_1 (\\Delta x_1)^2 = \\frac{1}{2} (k_1 \\Delta x_1) \\Delta x_1 = \\frac{1}{2} |\\vec{F}| \\Delta x_1$

$latex U_2 = \\frac{1}{2} k_2 (\\Delta x_2)^2 = \\frac{1}{2} (k_2 \\Delta x_2) \\Delta x_2 = \\frac{1}{2} |\\vec{F}| \\Delta x_2$

$latex \\text{Since } \\Delta x_1 < \\Delta x_2, \\text{ then } U_1 < U_2$

(A) $latex \\text{Extension: } \\Delta x_1 < \\Delta x_2 \\text{, Stored Potential Energy } U_1 < U_2$