Answer:
-0.23694 N
Explanation:
g = Acceleration due to gravity = 9.81 m/sĀ²
m = Mass of the Earth = Ā 5.972 Ć 10Ā²ā“ kg
G = Gravitational constant = 6.67 Ć 10ā»Ā¹Ā¹ mĀ³/kgsĀ²
r = Radius of Earth = 6371000 m
dr = Height = 1 mile = 1609.34 m
Acceleration is given by
[tex]a=\dfrac{GM}{r^2}[/tex]
Change in acceleration is given by
[tex]da=-2\dfrac{GM}{r^3}dr[/tex]
[tex]w=ma\\\Rightarrow w=m\dfrac{GM}{r^2}\\\Rightarrow w=469\ N[/tex]
[tex]dw=mda\\\Rightarrow dw=-m2\dfrac{GM}{r^3}dr\\\Rightarrow dw=-2w\dfrac{dr}{r}\\\Rightarrow dw=-2\times 469\times \dfrac{1609.34}{6.371\times 10^{6}}\\\Rightarrow dw=-0.23694\ N[/tex]
The change in weight is -0.23694 N
The change in your weight if you were to ride an elevator from the street level where you weigh 469N to the top of the building is; -0.237 N
The formula for acceleration here is;
a = GM/rĀ²
Where;
G is gravitational constant = 6.67 Ć 10ā»Ā¹Ā¹ mĀ³/kg.sĀ²
M is mass of earth = 5.972 Ć 10Ā²ā“ kg
r is distance from center of earth
Since we are trying to find change in weight, let us first find the change in acceleration with respect to r;
da/dr = -2GM/rĀ³
da = Ā -(2GM/rĀ³) dr
Thus, change in weight from top to bottom is;
W_top - W_bottom = m(da)
Now, weight at bottom is gotten from the formula;
W_bottom = GmM/rĀ²
Also, W_bottom = m(da) since we are dealing with change in weight.
Thus;
m(da)= -(2GmM/rĀ³) dr
Recall that GmM/rĀ². Thus;
m(da) = -2W_bottom Ć dr/r
where;
W_bottom = 469 N
r is radius of earth = 6371000 m
dr = 1 mile = 1609.34 m
Thus;
m(da) = -2 Ć 469 Ć 1609.34/6371000
m(da) = -0.237 N
Read more about Newton's law of gravitation at; https://brainly.com/question/14166269
