Sign Convention for Spring and Pendulum Equations of Motion

[gomerpyle]

If a mass was hanging vertically from a spring under its own weight.

In the static equilibrium position, using up as positive, the spring force is +kx and gravity is -mg. Add them up to get: kx - mg = 0. Using down as positive, the spring force is -kx and gravity is +mg. Add them: -kx + mg = 0.

In this case it does not matter which convention we use to determine that the initial static deflection is canceled out by the weight, but when we stretch the spring and derive the equation of motion we have either:

mx" + kx = 0 (down taken as positive)
mx" - kx = 0 (up taken as positive)

If we solved the second equation, it would not correctly describe the motion of the system. However, without knowing this how would someone know which sign convention to use? Is there a rule of thumb that would suggest downward as positive would be correct?

The same thing happens with a simple pendulum.

The equation of motion ends up being θ" + g/L*sinθ = 0
However, like with the example above this depends on which direction you take to be positive or negative for the mgsinθ term to be acting in. But, without knowing that you should expect an oscillatory response how would you you know that θ" - g/L*sinθ = 0 is incorrect?

 

[tiny-tim]

hi gomerpyle!

mx" + kx = 0 (down taken as positive)
mx" - kx = 0 (up taken as positive)

no, you're kidding yourself

the second one uses the same symbol for two different things

we can rewrite them as …

mDown" + kDown = 0 (down taken as positive)
mDown" - kUp = 0 (up taken as positive) …​

both correct!