91Ó°ÊÓ

When the spring is allowed to oscillate, this potential energy is released. When the spring returns to its equilibrium, all energy is converted to kinetic energy, and the maximum speed is achieved.

The general equation for the period of the oscillator is written by,

$$\begin{gathered} T=\frac{2\mathrm{Ï€}}{\mathrm{Ó\neg }} \\ \mathrm{Ó\neg }=\sqrt{\frac{k}{m}} \\ T=2\mathrm{Ï€}\sqrt{\frac{\mathrm{m}}{\mathrm{k}}} \end{gathered}$$

Where $T$is period, $\mathrm{Ó\neg }$is the angular frequency, $k$is the stiffness of spring, and$m$ is the mass of the block.

Part a)

If we double the mass but keep the stiffness the same, then the period is

$$\begin{gathered} T\text{'}=2\mathrm{π}\sqrt{\frac{2\mathrm{m}}{\mathrm{k}}} \\ \mathrm{T}\text{'}=\sqrt{2×2\mathrm{π}\sqrt{\frac{\mathrm{m}}{\mathrm{k}}}} \\ \mathrm{T}\text{'}=\sqrt{2}\mathrm{T} \end{gathered}$$

Compare to $T\text{'}=\sqrt{2}T$

$b=\sqrt{2}$

Thus, if we double the mass but keep the stiffness the same, the period change by the factor $\sqrt{2}$.

Part b)

If we double the spring stiffness but keep the mass the same, then the period is

$$\begin{gathered} T\text{'}=2\mathrm{π}\sqrt{\frac{\mathrm{m}}{2\mathrm{k}}} \\ \mathrm{T}\text{'}=\frac{1}{\sqrt{2}}×2\mathrm{π}\sqrt{\frac{\mathrm{m}}{\mathrm{k}}} \\ \mathrm{T}\text{'}=\frac{\mathrm{T}}{\sqrt{2}} \end{gathered}$$

Compare to $T\text{'}=bT$

$b=\frac{1}{\sqrt{2}}$

Thus,if we double the spring stiffness but keep the mass the same, the period change by the factor$\frac{1}{\sqrt{2}}$ .

Part c)

If we double the mass and also double the spring stiffness, then the period is

$$\begin{gathered} T\text{'}=2\mathrm{Ï€}\sqrt{\frac{2\mathrm{m}}{2\mathrm{k}}} \\ \mathrm{T}\text{'}=2\mathrm{Ï€}\sqrt{\frac{\mathrm{m}}{\mathrm{k}}} \\ \mathrm{T}\text{'}=\mathrm{T} \end{gathered}$$

Compare to $T\text{'}=bT$

$b=1$

Thus,if we double the mass and also double the spring stiffness, the factor is 1.

Part d)

If we double the amplitude, then the period will be the same, so that the period change by the factor is 1.