# Problem: Near the top of the Citigroup Center building in New York City, there is an object with mass of 4.8 × 105 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, in N/m, should the springs have to make them oscillate with a period of 1.2 s?(b) What energy, in joules, is stored in the springs for a 1.6 m displacement from equilibrium ?

###### FREE Expert Solution

The period of oscillation:

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

Energy stored in a spring:

$\overline{){\mathbf{E}}{\mathbf{=}}\frac{\mathbf{1}}{\mathbf{2}}{\mathbf{k}}{{\mathbf{x}}}^{{\mathbf{2}}}}$

Part (a)

From the expression of the period of oscillation, the force constant, k is expressed as:

$\mathbit{k}\mathbf{=}\frac{\mathbf{4}{\mathbf{\pi }}^{\mathbf{2}}\mathbf{m}}{{\mathbf{T}}^{\mathbf{2}}}$

###### Problem Details

Near the top of the Citigroup Center building in New York City, there is an object with mass of 4.8 × 105 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, in N/m, should the springs have to make them oscillate with a period of 1.2 s?
(b) What energy, in joules, is stored in the springs for a 1.6 m displacement from equilibrium ?