Near the top of the Citigroup Bank building in New York City, there is a 4.00 105 kg mass on springs having 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 mass, which oscillates instead of the building.

(a) What effective force constant should the springs have to make the mass oscillate with a period of 3.00 s? N/m

(b) What energy is stored in the springs for a 2.00 m displacement from equilibrium?

(a) k = 1.76 * 10⁶ N/m

(b) E = 3.52 * 10⁶ J

Explanation:

(a)

The period (T) of a spring = 2π√ (m/k)

where m = mass of the spring in kg, k = spring constant.

T = 2π√ (m/k) ... equation 1

making k the subject of the equation,

k = 4π² (m) / T² ... equation 2

Where m = 4.00 * 10⁵ kg, T = 3.00 s, π = 3.143

Substituting these values into equation 2

k = 4 (3.143) ² (4.0*10⁵) / 3²

k = (1.58 * 10⁷) / 9

k = 1.76 * 10⁶ N/m

(b)

The energy stored (E) in a spring = 1/2ke²

Where k = spring constant, e = extension.

E = 1/2ke²

k = 1.76 * 10⁶ N/m, e = 2.00 m

∴E = 1/2 (1.76 * 10⁶) (2) ²

E = 2 * 1.76 * 10⁶

E = 3.52 * 10⁶ J