Question

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?

Answer 1

(a) k = 1.76× 10⁶ N/m

(b) E = 3.52 × 10⁶ J

Step-by-step 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