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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 oscillates 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 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?

Respuesta :

boffeemadrid

Answer:

13.00079 s

0.36246 m/s

Explanation:

L = Length of cable = 42 m

g = Acceleration due to gravity = 9.81 m/s²

[tex]\omega[/tex] = Angular frequency =[tex]2\pi f[/tex]

f = Frequency = [tex]f=\frac{1}{T}[/tex]

A = Ampliltude = 75 cm

Time period is given by

[tex]T=2\pi\sqrt{\frac{L}{g}}\\\Rightarrow T=2\pi\sqrt{\frac{42}{9.81}}\\\Rightarrow T=13.00079\ s[/tex]

The period of oscillation fo the building is 13.00079 s

As the motion is in simple harmonic motion we use the following equation to find maximum velocity

[tex]v_m=\omega A\\\Rightarrow v_m=2\pi fA\\\Rightarrow v_m=2\pi \frac{1}{13.00079}\times 0.75\\\Rightarrow v_m=0.36246\ m/s[/tex]

The maximum speed of the pendulum is 0.36246 m/s