Tall buildings are designed to sway in the wind. In a 100 km/h...

Question

Tall buildings are designed to sway in the wind. In a 100 km/h wind, suppose the top of a 110-story building oscillates horizontally with an amplitude of 15 cm at its natural frequency, which corresponds to a period of 7.0 s. Assuming SHM, find the maximum horizontal velocity and acceleration experienced by an employee as she sits working at her desk located on the top floor. Compare the maximum acceleration (as a percentage) with the acceleration due to gravity.

Accepted Answer

Hello, fellow physicists today, we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use. In order to solve this problem. A student places a ruler at the edge of a table. He holds one end of the ruler inside the table and makes the other end stick outside. Then he slaps the end of the ruler sticking out the table at the speed of 32 kilometers per hour. This makes the ruler oscillate in simple harmonic motion. Given that the part of the ruler sticking out of the table is 15 centimeters in length, the period oscillation is 0.78 seconds and the amplitude is seven centimeters determine the maximum velocity and acceleration of the ruler's oscillating end express the maximum acceleration as a percentage of the gravitational acceleration G. So that's our end goal as our end goal is we're ultimately trying to solve for two separate answers. Our first answer, we're trying to determine the maximum velocity. And then our second answer, we're trying to determine the maximum acceleration as a percentage of G awesome. So looking at our multiple choice answers. Our first answer will, in our multiple choice answers will be the maximum velocity value V max is equal to something and they're all gonna be in units of meters per second. And then our second answer will be our maximum acceleration as a percentage of G and they're all gonna be written as percentages. So let's read them off to see what our final answer pair might be. A is 0.56 and 46. B is 0.090 and 7.4 C is 1.2 and 99 and D is 8.9 and 730. So first off, let us note that we are given the period of oscillation, which we're gonna call capital T, which is equal to 0.78 seconds. We're also given the amplitude which we're gonna call capital A which is equal to seven centimeters, which is also equal to 0.07 m. So you can use dimensional analysis or you can quickly look up this conversion to do the conversion for this. We also need to recall at this point. We need to recall and use the angular velocity equation which states that the angular velocity omega is equal to two multiplied by pi divided by the period of oscillation, which is equal to two multiplied by pi divided by 0.78 seconds, which is equal to 8.055 when we round to three decimal places and its units will be radians per second for our angular velocity. Now, at this point, we need to recall and use the equation to solve for the maximum velocity which we will write as V Max and V max is equal to omega multiplied by capital A where capital A is our amplitude. And this is equal to two multiplied by pi divided by T multiplied by A and this is all equal to two multiplied by pi multiplied by A, all divided by T Fantastic. So now at this point, we can plug in all of our known variables to solve for the numerical value of V max. So we will get that V max is equal to two multiplied by pi multiplied by 0.07 m all divided by T which is 0.78 seconds, which is equal to 0.56 m per second when we round to two decimal places or to two significant figures. So now at this point, we need to recall and use the maximum acceleration equation which we're gonna call a max. So let us recall that the maximum acceleration equation states that omega squared multiplied by the amplitude is equal to our maximum acceleration. So this is equal to two multiplied by pi divided by T all squared multiplied by our area. So that will mean that A max is equal to four multiplied by pi squared multiplied by A, all divided by T squared. So now at this point, we can recall that we can find the maxim acceleration as a percentage of G by dividing A max by G and multiplying it by 100% like so, so putting what we just said in words into writing, so a max divided by G multiplied by 100 is equal to four multiplied by pi squared multiplied by a or more specifically capital, A divided by T divided by G multiplied by 100. And we can plug in all of our known variables to solve for our final answer numerically. So we'll take or multiplied by pi squared multiplied by 0.07 m. And we need to divide this value by 0.78 seconds. And we're gonna divide this value by the acceleration due to gravity, which as we should recall to be 9.81 meters per second squared. And then we need to multiply this value by 100. And when we plug that into our calculator and we round to the nearest whole number, you will get 46%. And that is our final answer. Hooray, we did it. So we found our two final answers because we found the max to be 0.56 m per second. And we found our maximum acceleration as a percentage of G to be 46%. So looking at our multiple choice answers, the correct answer has to be the letter A. Our maximum velocity is V max is equal to 0.56 m per second and our maximum acceleration as a percentage of G is equal to 46%. Thank you so much for watching. Hopefully that helped and I can't wait to see in the next video. Bye.

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