Somebody's Wrong

Two students, Jon and Javin, are trying to find the spring constant $k$ of an ideal, massless spring.

Both of their setups are the same:

1. Hang the spring in a vertical position and measure the length of the unextended spring $L_1$ .
2. Hang a known mass of mass $m$ at the bottom of the spring, and measure the length of the extended spring $L_2$
3. Find the length of extension: $x = L_2 - L_1$ .

However, Jon and Javin have different ways of finding $k$ .

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Jon proposes:

Suppose the mass is hung on the spring and supported such that the spring remains unextended. This mass has more gravitational potential energy compared to when the mass is unsupported and the spring extends to length $L_2$ . In fact, it has $mgx$ more gravitational potential energy, where $g$ is the gravitational field strength. This additional gravitational potential energy must be converted to potential energy of the spring as the spring extends to length $L_2$ , which is $\displaystyle \frac{1}{2} kx^2$ by Hooke's Law. Hence by Conservation of Energy, $\displaystyle mgx = \frac{1}{2}kx^2$ , so $\displaystyle k=\frac{2mg}{x}$ .

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Javin proposes:

The gravitational force on the mass is $mg$ , where $g$ is the gravitational field strength, while the force on the mass by the spring is $kx$ by Hooke's Law. Since the mass is in equilibrium, by Newton's First Law, the forces on the mass must balance: $kx = mg$ . Hence $\displaystyle k = \frac{mg}{x}$ .

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Jon's and Javin's expressions for $k$ differ. Who has the correct reasoning?