Question

Mile-high building.In 1956, Frank Lloyd Wright proposed the construction of a mile-high building in Chicago. Suppose the building had been constructed. Ignoring Earth’s rotation, find the change in your weight if you were to ride an elevator from the street level, where you weigh$\mathbf{600}\mathbf{N}$, to the top of the building.

Short answer

Change in weight from bottom to top of the building is$-0.303N$

Step-by-step solution

The given data

Weight at street level is$W=600N$ (ignoring Earth’s rotation)

Understanding the concept of Gravitational acceleration

We use the concept of gravitational acceleration to find the change in weight of the person from the bottom to the top of the building. We can differentiate the equation to find the change in the acceleration. Using the change in acceleration, we can find a weight change.

Formula:

Gravitational acceleration,$a_g=\frac{GM}{r^2}$ (i)

Calculation of change in weight from top to bottom

Differentiating acceleration of equation (i), we get

$d{a}_g=-2\frac{G{M}_e}{r^3}dr$

So the change in weight is,

$W_{top}-W_{bottom}\approx m\left(d{a}_g\right)$

We know,

$W_{bottom}=\frac{G{M}_{e}m}{R^2}$

We get,

$$\begin{gathered} md{a}_g=-\frac{2Gm{M}_e}{R^3}dr \\ =-\frac{2W_{bottom}dr}{R} \end{gathered}$$

Putting the values we get,

$$\begin{gathered} md{a}_g=-\frac{2\left(600\right)(1.61\times {10}^3)}{6.37\times {10}^6} \\ =-0.303N \end{gathered}$$

The weight change is$-0.303N$

Negative sign indicates that decrease in weight.