91Ó°ÊÓ

1. The compression of the spring is,$x=5.5\mathrm{cm}=0.055\mathrm{m}$
2. The mass of the cork is,$m=3.8\mathrm{g}=3.8×{10}^{-3}\mathrm{kg}$
3. The spring is stretched through a distance of $d=1.5\mathrm{cm}=1.5×{10}^{-2}\mathrm{m}$

Use the concept of energy conservation law and elastic potential energy of the spring. According to the law of energy conservation, energy can neither be created, nor be destroyed.

Formulae:

$$\begin{gathered} U\left(x\right)=\frac{1}{2}k{x}^2 \\ K=\frac{1}{2}m{v}^2 \end{gathered}$$

where, K is kinetic energy, U(x)is potential energy, m is mass, v is velocity, x is displacement and kis spring constant.

From the slope of the graph, find the value of the force constant,

$$\begin{gathered} k=\frac{∆F}{∆x} \\ k=\frac{0.4-0.2}{4-2} \\ k=0.10\mathrm{N}/\mathrm{cm} \\ k=10\mathrm{N}/\mathrm{m} \end{gathered}$$

Now, the speed of the cork when the spring passes through its relaxed position can be find as follow:

When the cork is released, the potential energy of the compressed spring is converted to kinetic energy of the cork.

$$\begin{gathered} \frac{1}{2}k{x}^2=\frac{1}{2}m{v}^2 \\ {v}^2=\frac{k{x}^2}{m} \\ v=\sqrt{\frac{k}{m}}x \\ v=\sqrt{\frac{10\mathrm{N}/\mathrm{m}}{0.0038\mathrm{kg}}}×0.055\mathrm{m} \\ v=2.8\mathrm{m}/\mathrm{s} \end{gathered}$$

Hence, the speed of the cork when the spring passes through its relaxed position is, $v=2.8\mathrm{m}/\mathrm{s}$.

In this case, the spring stretches by d. Hence, potential energy of the compressed spring is converted into kinetic energy and potential energy of the cork,

$$\begin{gathered} \frac{1}{2}k{x}^2=\frac{1}{2}m{v}^2+\frac{1}{2}k{d}^2 \\ \frac{1}{2}m{v}^2=\frac{1}{2}k{x}^2-\frac{1}{2}k{d}^2 \\ m{v}^2=k\left(x^2-d^2\right) \\ v=\sqrt{\frac{k\left(x^2-d^2\right)}{m}} \\ v=\sqrt{\frac{10\mathrm{N}/\mathrm{m}\left[{\left(0.055\mathrm{m}\right)}^2-{\left(0.015\mathrm{m}\right)}^2\right]}{3.8×{10}^{-3}\mathrm{kg}}} \\ v=2.7\mathrm{m}/\mathrm{s} \end{gathered}$$

Hence, the speed of the cork at the time of release is, $v=2.7\mathrm{m}/\mathrm{s}$.

Therefore, thevelocity of the cork can be found by using the conservation of energy.