4. The monument commemorating the Battle of San Jacinto in Texas stands almost [tex]2.00 \times 10 \, \text{m}[/tex] and is topped by a [tex]2.00 \times 10^5 \, \text{kg}[/tex] star. Imagine that a [tex]2.00 \times 10^5 \, \text{kg}[/tex] mass is placed on a spring platform. The platform requires 0.80 s to oscillate from the top to the bottom positions.

What is the spring constant of the spring supporting the platform?

The spring constant 'k' can be calculated using the formula k = (4π²m) / T², where m is the mass and T is the full oscillation period. Given the half oscillation period of 0.80s, the full period is 1.60s and the mass is 2.00 x 10&sup5; kg.

To find the spring constant of the spring supporting the platform, we can use the formula for the period of a mass-spring system in simple harmonic motion, T = 2π√(m/k), where T is the period, m is the mass, and k is the spring constant. Given that the period (the time to oscillate from the top to the bottom positions and back to the top) is 0.80 s for one half oscillation, the full period is 1.60 s. Hence, we can rearrange the formula to solve for k:

k = (4π²m) / T²

Plugging in the values, we have:

k = (4π² × 2.00 × 10&sup5; kg) / (1.60 s)²

Thus, we can calculate the spring constant k for the given mass and oscillation period.