The given problem can be exemplified in the following diagram:
To determine the constant of the spring we can use Hook's law, which is the following:
[tex]F=k\Delta x[/tex]Where:
[tex]\begin{gathered} F=\text{ force on the string} \\ k=\text{ string constant} \\ \Delta x=\text{ difference in length} \end{gathered}[/tex]Now, we solve for "k" by dividing both sides by the difference in length:
[tex]\frac{F}{\Delta x}=k[/tex]The force on the string is equivalent to the weight attached to it. The weight is given by:
[tex]W=mg[/tex]Where:
[tex]\begin{gathered} W=\text{ weight} \\ m=\text{ mass} \\ g=\text{ acceleration of gravity} \end{gathered}[/tex]Substituting in the formula for the constant of the spring we get:
[tex]\frac{mg}{\Delta x}=k[/tex]Now, we substitute the values:
[tex]\frac{(3kg)(9.8\frac{m}{s^2})}{35\operatorname{cm}-30\operatorname{cm}}=k[/tex]Before solving we need to convert the centimeters into meters. To do that we use the following conversion factor:
[tex]100\operatorname{cm}=1m[/tex]Therefore, we get:
[tex]\begin{gathered} 35\operatorname{cm}\times\frac{1m}{100\operatorname{cm}}=0.35m \\ \\ 30\operatorname{cm}\times\frac{1m}{100\operatorname{cm}}=0.30m \end{gathered}[/tex]Substituting in the formula we get:
[tex]\frac{(3kg)(9.8\frac{m}{s^2})}{0.35m-0.30m}=k[/tex]Solving the operations:
[tex]588\frac{N}{m}=k[/tex]Therefore, the constant of the spring is 588 N/m.
