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A simple spring has length l and force constant k. It is cut into two springs of length l1 and l2 such that l1=nl2( n is an integer) the force constant of a spring of length l1 is
A. k(1+n)
B. nkn+1
C. (1+n)kn
D. k

Answer
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Hint: We know that the length of the spring and force constant are inversely related. Total length l of the spring is given as a sum of lengths l1 and l2. By substituting for each length in terms of respective force constants and on solving we can find the value of the spring constant of length l1 in terms of the total spring constant k.


Complete step by step answer:
It is given that a spring has a length of l .
The force constant of the spring is k.
Then the spring is cut into two springs of length l1 and l2 .
Relationship between l1 and l2 is given as
l1=nl2
Where, n is an integer.
We need to find the force constant of the spring of length l1 .
We know that the length of a spring and force constant are inversely related.
So, we can write it as
l1k
l=ck (1)
Where, c is a constant of proportionality.
Let k1 be the force constant of spring with length l1 and k2 be the force constant of spring with length l2.
Then we can write
l1=ck1 (2)
And
l2=ck2 (3)
We know that the total length l is the sum of l1 and l2 .
l=l1+l2
l=l1+l1n
l1=nl2
Substituting the value of l and l1from equation 1 and 2, we get
ck=ck1+cnk1
Now let us solve for k1 .
1k=1k1+1nk1
1k=n+1nk1
k1=n+1nk
This is the value of the spring constant of length l1 .
Hence, the correct answer is option C.

Note: Remember that the length and spring constant are inversely related. If we increase the length of the spring then the force constant will decrease and if you decrease the length of spring then the force constant will increase. So, if we cut a spring in half the spring constant of each half will be doubled. In our case since l1=nl2 The relation between spring constants of these parts will then be k2=nk1
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