Question

Two identically-charged particles are fastened to the two ends of a spring of spring constant 100 N m−1 and natural length 10 cm. The system rests on a smooth horizontal table. If the charge on each particle is 2.0 × 10−8 C, find the extension in the length of the spring. Assume that the extension is small as compared to the natural length. Justify this assumption after you solve the problem.

Answer 1

The extension in the length of the spring $36 \times 10^{-5} \mathrm{m}\end$ .

Explanation:

Given data in the question :

$\begin{equation}\mathrm{r}=10 \mathrm{cm}=10^{-1} \mathrm{m}\end$

$\begin{equation}q=2.0 \times 10^{-8} c\end$

$\begin{equation}F=\frac{k q_{1} q_{2}}{r^{2}}\end$

$\begin{equation}F=\frac{9 \times 10^{9} \times 2 \times 10^{-8} \times 2 \times 10^{-8}}{0.1^{2}}\end$

$\begin{equation}F=\frac{36 \times 10^{9} \times 10^{-8} \times 10^{-8}}{10^{-2}}\end$

$\begin{equation}F=\frac{36 \times 10^{-7}}{10^{-2}}\end$

$\begin{equation}F=36 \times 10^{-5} \mathrm{m}\end$

Thus, the length of the spring is $\begin{equation}F=36 \times 10^{-5} \mathrm{m}\end$.

Justification :

Yes, that is a reasonable assumption.

• Because at the ends of the spring there are two identical charges and they exert repulsive force on each other.
• In the spring, an extension x is created because of the repulsive force between the charges.
• The fountains are made of elastic material.
• When a spring is extended then a restore force acts on it that is always proportional to the extent produced and directed contrary to the direction of the force applied.
• The restocking force depends on the material's elasticity.
• If the extension is small then only the restore force to the extension is proportionally the extensions comparable to the natural length of the spring then the restore force will depend on the increased powers of the produced extension.

Answer 2

Answer:

Let us take the case when the extension is maximum. Here, the repulsion between charges is balanced by spring force and x is the extension of spring.

Then the answer is obtained by solving the cubic equation.

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But solving a cubic equation is decimals is difficult. So we can simplify matter by taking the initial assumption as in the 3rd image, ie by assuming that the extension x is small as compared to natural length.