Why is a Shorter Spring Stiffer Than a Longer One?

[sweetrose]

a physics question i was assigned was:

If two springs are made of the exact same material, but one is shorter than the other, why is the shorter spring stiffer than the other spring?

I know it is related to Hooke's law being that if a spring has a higher K value, it is stiffer, but I don't understand why the shorter spring's coils wouldn't stretch the same amount as the other spring's coils if the downward force on each spring is the same. So basically, I want to understand why the shorter spring has a higher K value than the longer spring.

 

[HallsofIvy]

What reason do you have for believing that "a shorter spring is stiffer"?

 

[ShawnD]

It relates to strain - the ratio of elongation to original length. instead of putting it in terms of spring constant, put it in terms of modulus of elasticity (similar but not the same). I'll only use the terms that relate to what you are trying to figure out.

here is the formula for elasticity (NOT the same as spring constant)

elasticity = stress/strain

rearrange it once to get stress

stress = (elasticity)(strain)

Since elasticity is the same for both a long and short spring, I should just leave it out. Stress is actually pressure but since both springs would have the same area, I'll just leave out the area part and make it a force.

F = strain

So now you know the force is dependent on strain, so what is strain? Strain is the change in length divided by the original length.

strain = \frac{\DELTA L}{L_o}

Shorter springs have higher a higher strain for the same amount of elongation and since F (force) is affected by strain, the k value must increase.