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Taipei 101 (a 101-story building in Taiwan) is sited in an area that is prone to earthquakes and typhoons, both of which can lead to dangerous oscillations of the building. To reduce the maximum amplitude, the building has a tuned mass damper, a 660,000 kg mass suspended from 42 m long cables that oscillate at the same natural frequency as the building. When the building sways, the pendulum swings, reaching an amplitude of 75 cm in strong winds or tremors. Damping the motion of the mass reduces the maximum amplitude of the oscillation of the building.a. What is the period of oscillation of the building?b. During strong winds, how fast is the pendulum moving when it passes through the equilibrium position?

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cryssatemp

Answers:

a) 13 s

b) 0.362 m/s

Explanation:

We have the following data:

[tex]m=660000 kg[/tex] is the mass of the mass damper

[tex]L=42 m[/tex] is the length of the pendulum

[tex]A=75 cm \frac{1m}{100 cm}=0.75 m[/tex] is the amplitude

a) Period of oscillation:

This can be solved by the following equation:

[tex]T=2 \pi \sqrt{\frac{L}{g}}[/tex] (1)

Where:

[tex]T[/tex] is the period

[tex]g=9.8 m/s^{2}[/tex] is the acceleration due gravity

[tex]T=2 \pi \sqrt{\frac{42 m}{9.8 m/s^{2}}}[/tex] (2)

[tex]T=13 s[/tex] (3)

b) Maximum Velocity:

The velocity in a pendulum is maximum [tex]V_{max}[/tex] when the pendulum is in its mean position and the amplitude is maximum. So, the equation in this case is:

[tex]V_{max}=A \frac{2 \pi}{T}[/tex] (4)

[tex]V_{max}=0.75 m \frac{2 \pi}{13 s}[/tex] (5)

[tex]V_{max}=0.362 m/s[/tex]

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