A nonlinear spring generates a force $f = \beta x ^ { 2 }$, where $\beta$ is a constant and $x$ is the distance that the spring is compressed or stretched. The spring hangs from the ceiling, and a mass $m$ is attached to the other end of the spring. The distance that the spring is stretched is
A) $\beta / ( m g )$
B) $\sqrt { m g / \beta }$
C) $\beta \sqrt { m g }$
D) $\sqrt { \beta / m / g }$

Question

Quizplus · Accepted Answer

The force generated by the spring must balance the gravitational force acting on the mass, so $f = mg$. Substituting $f = \beta x^2$ gives $\beta x^2 = mg$, solving for $x$ gives $x = \sqrt{mg/\beta}$.

A Nonlinear Spring Generates a Force f=βx2f = \beta x ^ { 2 }f=βx2

A Nonlinear Spring Generates a Force $f = \beta x ^ { 2 }$