To find the value of $k$, we can rearrange the formula given:

$T = 2\pi \sqrt{\frac{m}{k}}$

Given:

• $T = 3.70$
• $m = 67$

First, isolate $\sqrt{\frac{m}{k}}$:

$\frac{T}{2\pi} = \sqrt{\frac{m}{k}}$

Square both sides to remove the square root:

$\left(\frac{T}{2\pi}\right)^2 = \frac{m}{k}$

Rearrange to solve for $k$:

$k = \frac{m}{\left(\frac{T}{2\pi}\right)^2}$

Substitute the given values:

$k = \frac{67}{\left(\frac{3.70}{2\pi}\right)^2}$

Calculate the denominator:

$\frac{3.70}{2\pi} = \frac{3.70}{6.2832} \approx 0.5887$

Square this value:

$(0.5887)^2 \approx 0.3466$

Finally, compute $k$:

$k = \frac{67}{0.3466} \approx 193.26$

So, the value of $k$ is approximately $193.26 \, \text{N/m}$.

Would you like further details or have any questions about this calculation?

Here are 5 related questions to expand your understanding:

1. How does the period $T$ change if the mass $m$ is doubled?
2. What is the physical interpretation of the constant $k$ in this context?
3. How would the calculation change if $T$ was given in seconds instead of in terms of $2\pi$?
4. Can you derive the formula for $T$ from the principles of simple harmonic motion?
5. What units should $k$ have to be consistent with the given formula?

Tip: When dealing with formulas involving square roots, always ensure you isolate the square root term before squaring both sides to simplify the equation.