### Challenge Problems

A suspension bridge oscillates with an effective force constant of $1.00\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{8}\phantom{\rule{0.2em}{0ex}}\text{N/m}$. (a) How much energy is needed to make it oscillate with an amplitude of 0.100 m? (b) If soldiers march across the bridge with a cadence equal to the bridge’s natural frequency and impart $1.00\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{4}\phantom{\rule{0.2em}{0ex}}\text{J}$ of energy each second, how long does it take for the bridge’s oscillations to go from 0.100 m to 0.500 m amplitude.

Near the top of the Citigroup Center building in New York City, there is an object with mass of $4.00\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{5}\phantom{\rule{0.2em}{0ex}}\text{kg}$ on springs that have adjustable force constants. Its function is to dampen wind-driven oscillations of the building by oscillating at the same frequency as the building is being driven—the driving force is transferred to the object, which oscillates instead of the entire building. (a) What effective force constant should the springs have to make the object oscillate with a period of 2.00 s? (b) What energy is stored in the springs for a 2.00-m displacement from equilibrium?

Parcels of air (small volumes of air) in a stable atmosphere (where the temperature increases with height) can oscillate up and down, due to the restoring force provided by the buoyancy of the air parcel. The frequency of the oscillations are a measure of the stability of the atmosphere. Assuming that the acceleration of an air parcel can be modeled as $\frac{{\partial}^{2}{z}^{\prime}}{\partial {t}^{2}}=\frac{g}{{\rho}_{o}}\frac{\partial \rho \left(z\right)}{\partial z}{z}^{\prime}$, prove that ${z}^{\prime}={z}_{0}{}^{\prime}{e}^{t\sqrt{\text{\u2212}{N}^{2}}}$ is a solution, where *N* is known as the Brunt-Väisälä frequency. Note that in a stable atmosphere, the density decreases with height and parcel oscillates up and down.

Consider the van der Waals potential $U\left(r\right)={U}_{o}\left[{\left(\frac{{R}_{o}}{r}\right)}^{12}-2{\left(\frac{{R}_{o}}{r}\right)}^{6}\right]$, used to model the potential energy function of two molecules, where the minimum potential is at $r={R}_{o}$. Find the force as a function of *r*. Consider a small displacement $r={R}_{o}+{r}^{\prime}$ and use the binomial theorem:

${\left(1+x\right)}^{n}=1+nx+\frac{n\left(n-1\right)}{2!}{x}^{2}+\frac{n\left(n-1\right)\left(n-2\right)}{3!}{x}^{3}+\cdots $,

to show that the force does approximate a Hooke’s law force.

Suppose the length of a clock’s pendulum is changed by 1.000%, exactly at noon one day. What time will the clock read 24.00 hours later, assuming it the pendulum has kept perfect time before the change? Note that there are two answers, and perform the calculation to four-digit precision.

(a) The springs of a pickup truck act like a single spring with a force constant of $1.30\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{5}\phantom{\rule{0.2em}{0ex}}\text{N/m}$. By how much will the truck be depressed by its maximum load of 1000 kg? (b) If the pickup truck has four identical springs, what is the force constant of each?