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Oscillations — Physics STD 11 Science — Question

Rajasthan BoardEnglish MediumSTD 11 SciencePhysicsOscillations5 Marks
Question
A uniform spring whose unstretched length is l has a force constant k. The spring is cut into two pieces of unstretched lengths. l1, and l2 where l1 = nl2, and n is an integer. What are the corresponding force constants k1 and k1 in terms of n and k?
✓

Answer

Here l = l1 + l2 ...(i)

and l1 = nl2 ...(ii)

We khow, $\text{k}=\frac{\text{Mg}}{\text{l}}\cdots\text{(iii)}$

$\therefore\text{k}_1=\frac{\text{Mg}}{\text{l}_1}\cdots\text{(iv)}$

$\text{k}_2=\frac{\text{Mg}}{\text{l}_2}\cdots\text{(v)}$

Dividing equation (iv) by equation (iii) we find

$\frac{\text{k}_1}{\text{k}}=\frac{\text{l}}{\text{l}_1}=\frac{\text{l}_1+\text{l}_2}{\text{l}_1}=1+\frac{\text{l}_2}{\text{l}_1}$

From equation (ii) we find

$\frac{\text{l}_1}{\text{l}_2}=\text{n}$

$\therefore\frac{\text{k}_1}{\text{k}}=1+\frac{1}{\text{n}}$

$\text{k}_1=\Big(\frac{\text{n}+1}{\text{n}}\Big)\text{k}$

From equation (ii) and (iii), we find:

$\frac{\text{k}_2}{\text{k}}=\frac{\text{l}}{\text{l}_2}=\frac{\text{l}_1+\text{l}_2}{\text{l}_2}=\frac{\text{l}_1}{\text{l}_2}+1$

From equation (ii) we have $\frac{\text{l}_1}{\text{l}_2}=\text{n}$.

$\therefore\frac{\text{k}_2}{\text{k}}=(\text{n}+1)$

$\therefore\text{k}_2=\text{k}(\text{n}+1)$

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