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Step-by-Step Solution
Step 1: Identify the Original Spring Constant and the Length Ratio
We have an original spring with force constant $k$. It is cut into three segments whose lengths are in the ratio 1 : 2 : 3. This means if the total length is $l$, the three segments have lengths
$ \frac{l}{6}, \frac{2l}{6}, \frac{3l}{6} $ respectively.
Step 2: Calculate the Force Constants of Each Segment
The force constant $k_i$ of a spring segment is inversely proportional to its length for the same material and thickness. Specifically,
$ k_i = \frac{kl}{l_i} $.
Using this for each segment:
For length $l_1 = \frac{l}{6}$:
$ k_1 = \frac{kl}{l/6} = 6k $.
For length $l_2 = \frac{2l}{6}$:
$ k_2 = \frac{kl}{2l/6} = 3k $.
For length $l_3 = \frac{3l}{6}$:
$ k_3 = \frac{kl}{3l/6} = 2k $.
Step 3: Combine the Segments in Series
When springs are connected in series, their effective force constant $k'$ is given by the reciprocal sum of individual force constants:
$ \frac{1}{k'} = \frac{1}{k_1} + \frac{1}{k_2} + \frac{1}{k_3} $.
Substitute $k_1 = 6k$, $k_2 = 3k$, and $k_3 = 2k$:
$ \frac{1}{k'} = \frac{1}{6k} + \frac{1}{3k} + \frac{1}{2k} = \frac{1 + 2 + 3}{6k} = \frac{6}{6k} = \frac{1}{k} $.
Therefore,
$ k' = k $.
Step 4: Combine the Segments in Parallel
When springs are connected in parallel, their effective force constant $k''$ is the sum of the individual force constants:
$ k'' = k_1 + k_2 + k_3 = 6k + 3k + 2k = 11k $.
Step 5: Find the Ratio $k' : k''$
The ratio of the two effective force constants is:
$ \frac{k'}{k''} = \frac{k}{11k} = \frac{1}{11}. $
Hence,
$ k' : k'' = 1 : 11 $.
