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Hookes lawby joej24
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#1
Sep811, 10:13 PM

P: 74

1. The problem statement, all variables and given/known data
Whether or not the spring is vertical or not, assuming that the spring is ideal, is F_{net} always equal to kx? Is it incorrect to write the sum of the forces of a vertical spring as F_{net} = kx  mg? And does hooke's law account for the force of friction on a horizontal spring? If not would the sum of the forces look like this? F_{net} = kx  [itex]\mu[/itex]N 2. Relevant equations F = kx 3. The attempt at a solution 1. The problem statement, all variables and given/known data 2. Relevant equations 3. The attempt at a solution 


#2
Sep811, 10:25 PM

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#3
Sep811, 10:41 PM

P: 74

If a vertical spring is hanging and at rest, its net force = 0.
So that means F = 0 = kx  mg, but doesn't the force of the spring oppose the force of gravity? How does this make sense? The force of gravity always points downward ( is negative), but kx is negative when the spring is stretched and at rest. 


#4
Sep1011, 02:31 PM

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Hookes law
F = kΔy where Δy ≡ y  y_{0} In this expression, y_{0} is the position at which the spring is neither stretched nor compressed. The reason we can write F = ky is because, in general we can choose the origin of our coordinate system to be at y_{0} (i.e. we can choose y_{0} = 0) so that all other positions are measured from this reference point. But the key point here is that y is the spring displacement and as such, it is a vector. At the very least, for a 1D situation, y must have an intrinsic sign (+ or ) that tells you which direction this displacement is in. Next, we come to sign conventions. This is another case where we have the freedom to choose something. Gravity points downward. So, if you write F = mg, you've implicity chosen the sign convention that "downward is negative" (i.e. downward is the negative y direction). If so, this affects the sign of the displacement y. In the force balance equation, F = mg  ky, if you stretch the spring, then the position of the mass goes down below the position at which the spring is undisplaced. Since downward is negative, this means that the displacement is negative: y < 0. But if y is negative, then it follows that the spring force is positive: ky > 0. So, when the spring is stretched, the spring force is positive and points upward, counteracting gravity, just as you would expect. Similarly, if the spring is compressed, then the position, y, of the mass goes up above the position at which the spring is undisplaced. In other words the displacement is positive: y > 0. As a result, ky < 0 and a compressed spring has a restoring force that points downward, just as you would expect. You are free, of course, to choose the opposite sign convention in which "downward is positive". Under this sign convention, we write gravity as F = +mg, and the total force becomes: F = mg  ky Now, if the mass is initially at y = 0 (the position for which the spring is undisplaced) and then you stretch it by a small amount, its new position is physically below the zeropoint. Since downward has been defined to be positive, it follows that the displacement is positive: y > 0. Therefore ky < 0, and the spring force points in the negative direction (upward) just as one would expect. If the spring is compressed, the the mass moves above the zeropoint. Since upward is negative, it follows that the displacement is negative: y < 0. As a result, ky > 0, and the force for a compressed spring points in the positive (downward) direction as one would expect. This second sign convention is perhaps more intuitive, since a positive displacement in y corresponds to spring extension (lengthening) and a negative displacement in y corresponds to spring compression (shortening). 


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