## Near the top of the Citigroup Center building in New York City, there is an object with mass of 4.8 x 105 kg on springs that have adjustable

Question

Near the top of the Citigroup Center building in New York City, there is an object with mass of 4.8 x 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 X 50%
Part (a) What effective force constant, in N/m, should the springs have to make them oscillate with a period of 1.2 s? k = 9.5 * 106 9500000 X Attempts Remain 50%
Part (b) What energy, in joules, is stored in the springs for a 1.6 m displacement from equilibrium?

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1 year 2021-09-01T09:34:17+00:00 1 Answers 42 views 0

The force constant is  $$k =1.316 *10^{7} \ N/m$$

The energy stored in the spring is  $$E = 1.68 *10^{7} \ J$$

Explanation:

From the question we are told that

The mass of the object is  $$M = 4.8*10^{5} \ kg$$

The period is $$T = 1.2 \ s$$

The period of the spring oscillation is  mathematically represented as

$$T =2 \pi \sqrt{ \frac{M}{k}}$$

where  k is the force constant

So making k the subject

$$k = \frac{4 \pi ^2 M }{T^2}$$

substituting values

$$k = \frac{4 (3.142) ^2 (4.8 *10^{5}) }{(1.2)^2}$$

$$k =1.316 *10^{7} \ N/m$$

The energy stored in the spring is mathematically represented  as

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

Where x is the spring displacement which is given as

$$x = 1.6 \ m$$

substituting values

$$E = \frac{1}{2} (1.316 *10^{7}) (1.6)^2$$

$$E = 1.68 *10^{7} \ J$$