Universal law of gravitation ?

[warehouse54]

A "concept" question... I think the answer is yes... which means it probably is a definite "no."

The sun is directly below us at midnight, in line with the Earth's center. Are we then heavier at midnight, due to the sun's gravitational force on us, than we are at noon? Explain.

 

[Severian596]

The primary source of gravity that we feel here on Earth is earth. However other bodies affect our weight, such as the moon and the sun. The moon's gravity causes a very apparent effect on the oceans of earth; the tides.

During the rotation of the Earth you rotate with it in a circle. At one point you reach your farthest from the sun (which you called midnight) which means at noon you'd be closest to the sun. The vectors that represent your acceleration due to gravity (both Earth and sun) would look like this then:

Midnight

<----------------- earth
<- sun

Noon:
-----------------> earth
<- sun

Notice with the sun on the opposite side of you the Earth pulls you in the opposite direction. The MAGNITUDE of these arrows is way off scale, though. The force of gravity is divided by the square of the distance to the source of gravity as shown by:

http://en.wikipedia.org/math/b1365723a6484cc3ca9d0ce071b22f62.png

You are very close to the Earth compared to your distance to the sun, so the sun's gravitational force vector would be miniscule compared to the vector of the earth. But miniscule is not zero.

So yes, I argue that you are indeed heavier at night.

 

[Severian596]

For fun I quickly made a spreadsheet to do the calculations. I used the Newtonian equation for a 175 lb. person. I used one AU as the distance from the sun to the earth, and the radius of the Earth to find the person's distance to the sun at noon and midnight. This assumes the person is EXACTLY opposite the sun, standing on the Earth's plane of revolution, at both noon and midnight.

I represented the gravitational force values of the sun as (+) at midnight and (-) at noon, to represent their direction. Add the Sun's gravity to Earth's gravity to get a net "weight" for the person.

Here are my values:

Earth's gravity on you: 1.74781 E+20 N
Force (sun, midnight): +0.470795937 N
Force (sun, noon): -0.470876143 N

G: 6.67422 E-11 N*m^2/kg^2
M (sun): 1.98892 E+30 kg
M (earth): 5.9742 E+20 kg
M (person): 79.37866 kg
r (to sun, at mid): 149,604,371,000 m
r (to sun, at noon): 149,591,629,000 m

Distance from sun to earth: 1.49598E+11 m (1 Astronomical Unit)
Radius of earth: 6,371,000 m