# In a vertical spring, why is the amplitude $2mg/k$ and not $\sqrt{2mg/k} - mg/k$?

When the spring is in unstretched position, the extension is $$0$$. At the equilibrium, which is the mean position of the SHM, extension is $$\frac{mg}{k}$$.

The maximum extension possible should be, by conservation of Energy, $$mgx=\frac{1}{2}kx^2\quad x = \text{extension of spring}$$ and $$x$$ should come out to be $$\sqrt{\frac{2mg}{k}}$$.

Since the origin is at unstretched position and not the mean position,the amplitude should come out to be $$A = \sqrt{\frac{2mg}{k}} - \frac{mg}{k},$$ the displacement between mean position and extreme position.

The books I have seen have $$\frac{mg}{k}$$ as the amplitude. Doesn't it mean that the total extension of spring is $$\frac{2mg}{k}$$ (mean position + extreme position)? Won't that violate the law of conservation of mechanical energy, as it is greater than $$\sqrt{\frac{2mg}{k}}$$? How is this the amplitude, and what's the logic behind it?

• If $mg/k$ has the dimensions of a length, $\sqrt{2mg/k}$ cannot also have the dimensions of a length. Nov 20, 2018 at 6:14
• @G.Smith both are dimensionally consistent. Nov 20, 2018 at 6:49
• Got it man. Thanks . I was making a bad,bad,bad mistake. Nov 20, 2018 at 6:59
• I’m glad you found your mistake, but before you did you thought those two terms were dimensionally consistent. Do you understand that $X$ and $\sqrt{X}$ can be dimensionally consistent only if they are dimension-less, and then they can’t be a length? Nov 20, 2018 at 17:35
• @G.Smith Thanks, firstly. Cutting to the point,yes,I realise the facts you stated. High-school student like me sometimes go overboard and think dimensional analysis is for rookies(since it easy),and sometimes make realy ridiculous mistake while neglecting dimensions. Thanks again Nov 21, 2018 at 14:52

"and x should come out to be $$(\sqrt{\frac{2mg}{k}})$$".
It is wrong, by simple algebra; (divide both sides by x) it would be: $$({\frac{2mg}{k}})$$. Now you can see that it do not violate the law of conservation of mechanical energy!
This might help you... give you some intuition. Here, $$k$$ is the spring constant, NL is the normal length, EP is the equilibrium position, other symbols are having usual meaning.