In the study of Simple Harmonic Motion, one always come across questions on combine (usually 2) springs.
May I know what is the effective spring constants of 2 springs when they are:
a) joined in parallel,
b) joined in series?
Thanks.
Effective spring mass constant
a) k_1 + k_2
b) (k_1 k_2) / (k_1 + k_2)
This is pretty trivial to see for yourself actually :
a> In this case both springs will exert force, and if the displacement is x, then F = F_1 + F_2 = k_1x + k_2 x = (k_1 + k_2)x
b> In this case the strectch in both the sprngs is different, say x_1 & x_2, but the tension (or compression) in the two springs must match, and this is the force that will be passed on to the block.
Therefore: k_1 x_1 = k_2 x_2 = k' x;
& x_1 + x_2 = x
So: 1/k' = x/(k' x) = (x_1 + x_2)/ (k'x) = x_1/ (k' x) + x_2/(k' x) = x_1/ (k_1x_1) + x_2/(k_2 x_2) = 1/k_1 + 1/k_2
Hope that helps
b) (k_1 k_2) / (k_1 + k_2)
This is pretty trivial to see for yourself actually :
a> In this case both springs will exert force, and if the displacement is x, then F = F_1 + F_2 = k_1x + k_2 x = (k_1 + k_2)x
b> In this case the strectch in both the sprngs is different, say x_1 & x_2, but the tension (or compression) in the two springs must match, and this is the force that will be passed on to the block.
Therefore: k_1 x_1 = k_2 x_2 = k' x;
& x_1 + x_2 = x
So: 1/k' = x/(k' x) = (x_1 + x_2)/ (k'x) = x_1/ (k' x) + x_2/(k' x) = x_1/ (k_1x_1) + x_2/(k_2 x_2) = 1/k_1 + 1/k_2
Hope that helps

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 Joined: Mon Dec 18, 2006 3:04 am