The drawing (not to scale) shows one alignment of the sun, earth, and moon. The gravitational force Fsm that the sun exerts on the moon is perpendicular to the force Fem that the earth exerts on the moon. The masses are: mass of sun = 1.99 × 10^30 kg, mass of earth = 5.98 × 10^24 kg, mass of moon = 7.35 × 10^22 kg. The distances shown in the drawing are rSM = 1.50 × 10^11 m and rEM = 1.50 × 108 m. Determine the magnitude of the net gravitational force on the moon.
I've got an estimated answer as 1.4 x 10^21N, which is my net force.
use the universal gravitation formula to find the individual forces
add the two vectors to find the net force
the data looks like three sig fig ... so your answer should be also
when adding numbers written in sig figures, remember you can only add to the least sig figure in the largest number: ie 10200+34=10200, in that case a three sig figure plus a two sig figure yields a three sig. Example 2: 1020000+3433=102000, a three sig plus a four sig gives a three sig. So when adding vectors, watch this . Often we get bambluzoled when adding because of the cosine, sin conversion to get components makes us think each has more precison that it actually has.
To determine the magnitude of the net gravitational force on the moon, we can use the formula for the magnitude of the gravitational force:
F = G * (m1 * m2) / r^2
Where G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between the centers of the two objects.
Let's calculate the gravitational force between the sun and the moon first:
Fsm = G * (mSun * mMoon) / rSM^2
Using the given values:
G = 6.67 × 10^-11 N m^2 / kg^2
mSun = 1.99 × 10^30 kg
mMoon = 7.35 × 10^22 kg
rSM = 1.50 × 10^11 m
Plugging the values into the formula:
Fsm = (6.67 × 10^-11 N m^2 / kg^2) * ((1.99 × 10^30 kg) * (7.35 × 10^22 kg)) / (1.50 × 10^11 m)^2
Calculating this expression gives Fsm = 2.07867 × 10^20 N (rounded to five significant figures).
Now, let's calculate the gravitational force between the Earth and the Moon:
Fem = G * (mEarth * mMoon) / rEM^2
Using the given values:
mEarth = 5.98 × 10^24 kg
rEM = 1.50 × 10^8 m
Plugging the values into the formula:
Fem = (6.67 × 10^-11 N m^2 / kg^2) * ((5.98 × 10^24 kg) * (7.35 × 10^22 kg)) / (1.50 × 10^8 m)^2
Calculating this expression gives Fem = 1.166807333 × 10^26 N (rounded to five significant figures).
Now, since the gravitational force Fsm from the sun and the gravitational force Fem from the Earth are perpendicular, we can find the net gravitational force Fnet using the Pythagorean theorem:
Fnet = √(Fsm^2 + Fem^2)
Plugging in the calculated values:
Fnet = √((2.07867 × 10^20 N)^2 + (1.166807333 × 10^26 N)^2)
Calculating this expression gives Fnet = 1.166819311 × 10^26 N (rounded to five significant figures).
Therefore, the magnitude of the net gravitational force on the moon is approximately 1.167 × 10^26 N.