# Towers and skyscrapers appear to be rigid but they are designed to sway with the wind. Standing on the glass floor of the CN tower at 342 m above the ground the tower sways 40 cm from the vertical...

Towers and skyscrapers appear to be rigid but they are designed to sway with the wind. Standing on the glass floor of the CN tower at 342 m above the ground the tower sways 40 cm from the vertical during high winds. At t=3 s, the tower is 40 cm to the right of the vertical. The tower sways back to the vertical and, at t=9 s, it sways 40 cm to the left of the vertical.

a)Write the equation that models this motion of the building with respect to time

b) How would the equation in (a) be different if a person were standing in the Sky Pod at a height of 447m?

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This behavior can be modeled by a sinusoid (assuming that the building continues swaying in the same manner.)

We introduce some notation and a frame of reference: let y be the distance from the vertical at time t (t measured in seconds and y in cm.) We will take the distance to the right of vertical as positive, while the distance to the left of vertical we take to be negative.

The amplitude of the sinusoid is 40cm. (This is the maximum distance from the center.)

Since the building takes 6 seconds to sway from farthest right to farthest left (maximum to minimum), the period is 12 seconds. (The period is the time to complete one cycle.)

There is no vertical displacement (the building is "centered" at the vertical or 0cm left or right.)

Assuming we use the sine function there is no horizontal translation. (At time t=0 the building is vertical.)

The formula for a sinusoid is y=a[b*sin(t-h)]+k where a is the amplitude (the absolute value of the maximal distance from the center), b reflects the period (where `b=(2pi)/p ` with p the period), h is the horizontal translation and k the vertical translation. Substituting the values we get:

`y=40*sin((pi)/6 t) `

The graph:

Note that the graph of the model has the distance from the vertical at time t=0 at 0, t=3 at 40, and t=9 -40 as required.

(2) If we assume the Sky Pod also sways in the same manner, the period will remain roughly the same. (The side of the building is virtually rigid, so the building will reach the maximum position away from vertical at very close to the same time anywhere along the side of the building. There might be some flexing, but we ignore this.) The amplitude changes, however. Assuming the amount off of vertical is proportional, the maximum sway at can be found by solving ` 40/342=x/447 ==> x~~52.28"cm" `

Thus the new model will be `y=52.3 sin(pi/6 t) `