Question

Find the percentage change in acceleration due to gravity at a height of 32 km from the surface of the earth.

• A. $3\%$
• B. $5\%$
• C. $8\%$
• D. $1\%$

Answer 1

The correct option is D

$1\%$

Step 1: Given data

Height, $h=32km$

Formula used

Gravitate height, $g_h=g\left(1-\frac{2h}{R}\right)[whereh=height,r=radius]$

Step 2: Calculating percentage change in acceleration

We know that the radius of the earth is, $R_e=6.4\times {10}^6$

As we can see the height is very less than the radius of the earth

$h<<r$

We will use the formula

$$\begin{gathered} g_h=g\left(1-\frac{2h}{R}\right) \\ {g}_h=g\left(1-\frac{2\times 32km}{6400km}\right)[Ash=32km,r=6.4\times {10}^6] \\ {g}_h=g\left(\frac{99}{100}\right) \end{gathered}$$

Now we want a percentage change

$$\begin{gathered} \frac{g_h-g}{g}\times 100 \\ =\frac{g\left({\displaystyle \frac{99}{100}}\right)-g}{g}\times 100 \\ =\frac{g\left({\displaystyle \frac{99}{100}}-1\right)}{g}\times 100 \\ Aftersolvingweget, \\ Changeinacceleration=1\% \end{gathered}$$

Hence, option (D) is the correct answer.