Simple Harmonic Motion 4 Question 1
1. A spring whose unstretched length is $l$ has a force constant $k$. The spring is cut into two pieces of unstretched lengths $l _1$ and $l _2$ where, $l _1=n l _2$ and $n$ is an integer. The ratio $k _1 / k _2$ of the corresponding force constants $k _1$ and $k _2$ will be
(a) $n$
(b) $\frac{1}{n^{2}}$
(c) $\frac{1}{n}$
(d) $n^{2}$
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Answer:
Correct Answer: 1. (c)
Solution:
- If parameters like material, number of loops per unit length, area of cross-section, etc., are kept same, then force constant of spring is inversely proportional to its length.
In given case, all other parameters are same for both parts of spring.
$$ \begin{array}{rlrl} \text { So, } & k _1 & \propto \frac{1}{l _1} \text { and } k _2 \propto \frac{1}{l _2} \\ \therefore \quad & \frac{k _1}{k _2} & =\frac{l _2}{l _1} \\ & =\frac{l _2}{n l _2}=\frac{1}{n} \quad\left[\because l _1=n l _2\right] \end{array} $$
