Near the top of the Citigroup Center building in New York City, there is an object with a mass of...

Question:

Near the top of the Citigroup Center building in New York City, there is an object with a mass of {eq}4.2 \times 10^5 {/eq} 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 them oscillate with a period of 1.4 s in N/m?

b. What energy is stored in the springs for a 1.8 m displacement from equilibrium in J?

c. In terms of M, L, and x, what is the rod's moment of inertia I about the pivot point?

d. Calculate the rod's period T in seconds for small oscillations about its pivot point.

e. In terms of L, find an expression for the distance x{eq}_m {/eq} for which the period is a minimum.

Simple Harmonic Motion:

The ideal spring has a simple harmonic motion described with Hooke's law.

{eq}F_{applied}=k\cdot x {/eq}

K is considered the spring's constant

X is the displacement of an object from its equilibrium position.

Equilibrium is the position at which there is no net force acting a particle.

A spring that behaves according to Hooke's law is considered an ideal spring.

Simple harmonic motion occurs when the displacement and acceleration are related.

The acceleration is always directed towards the equilibrium point and is directly proportional to the distance.

Very common examples of simple harmonic motion are pendulums, Springs, and waves.

Answer and Explanation:

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Simple harmonic motion is present

We can use period of a mass-spring formula

{eq}T=2\pi \sqrt{\frac{m}{k}} {/eq}

Solve for k

{eq}k=m(\frac{2\pi}{...

See full answer below.


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Hooke's Law & the Spring Constant: Definition & Equation

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Chapter 4 / Lesson 19
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After watching this video, you will be able to explain what Hooke's Law is and use the equation for Hooke's Law to solve problems. A short quiz will follow.


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