Question

The Bernoulli Effect can have important consequences for the design of buildings. For example, wind can blow around a skyscraper at remarkably high speed, creating low pressure. The higher atmospheric pressure in the still air inside the buildings can cause windows to pop out. As originally constructed, the John Hancock Building in Boston popped windowpanes that fell many stories to the sidewalk below. (a) Suppose a horizontal wind blows with a speed of $\mathbf{11}\mathbf{.}\mathbf{2}\mathit{m}\mathbf{/}\mathit{s}$outside a large pane of plate glass with dimensions $\mathbf{4}\mathbf{.}\mathbf{00}\mathit{m}\mathbf{\times }\mathbf{1}\mathbf{.}\mathbf{50}\mathit{m}$. Assume the density of the air to be constant at $\mathbf{1}\mathbf{.}\mathbf{20}\mathit{k}\mathit{g}\mathbf{/}{\mathit{m}}^{\mathbf{3}}$. The air inside the building is at atmospheric pressure. What is the total force exerted by air on the windowpane? (b) What If? If a second skyscraper is built nearby, the airspeed can be especially high where wind passes through the narrow separation between the buildings. Solve part (a) again with a wind speed of $\mathbf{22}\mathbf{.}\mathbf{4}\mathit{m}\mathbf{/}\mathit{s}$, twice as high.

Short answer

(a) The total force exerted by air on the window plane is$F=451.584N$

(b) Twice the speed of air is twice the force$F_2=856.1N$

Step-by-step solution

Step 1:

According to Bernoulli’s principle force is given by:

$F=\frac{1}{2}\rho v^{2}A$

Step 2:

Part(a):

Given:

Speedofwind,$v=11.2m/s$

Dimensionofplate,$=4m\times 1.50m$

Density of air$\rho =1.2kg/m^3$

To find the total force exerted by air on the window plane,

$\begin{array}{rcl}F& =& \frac{1}{2}\rho v^{2}A\\ F& =& \frac{1}{2}\times 1.2\times {\left(11.2\right)}^2\times \left(4\times 1.50\right)\\ F& =& \frac{903.168}{2}\\ F& =& 451.584N\\ & & \end{array}$

Step 3:

Part(b):

Given:

Speedofwind,$v=22.4m/s$

From part (a) the total force is proportional to square of speed of air.

$\begin{array}{rcl}\frac{F_2}{{v_2}^2}& =& \frac{F_1}{{v_1}^2}\\ \frac{F_2}{{\left(22.4\right)}^2}& =& \frac{451}{{\left(11.2\right)}^2}\\ F_2& =& 856.1N\\ & & \end{array}$