Final answer:
Using the gravitational acceleration at the North Pole and Earth's polar radius, Earth's mass can be calculated to approximately 5.97 x 10^24 kg, matching closely with NASA's reported value of 5.9726 x 10^24 kg.
Step-by-step explanation:
To calculate Earth's mass using the acceleration due to gravity at the North Pole, we can use Newton's universal law of gravitation. This law states that the force due to gravity is proportional to the mass of the two objects and inversely proportional to the square of the distance between their centers. The formula we need to use here is:
F = G(m1*m2)/r^2
If we solve for m1 (which would be Earth's mass in this case), and assume m2 as the mass of a small object on Earth's surface where the force F is equal to the weight of the object (m2*g), we get:
m1 = (g*r^2)/G
where,
g is the acceleration due to gravity at the North Pole,
r is the radius of Earth at the pole, and
G is the gravitational constant (6.674 x 10^-11 N*(m/kg)^2).
Substituting the given values:
-
- g = 9.832 m/s^2
-
- r = 6356 km = 6356000 m
Earth's mass (m1) is then calculated to be:
m1 = (9.832 * (6356000)^2) / (6.674 x 10^-11)
Calculating this, we get Earth's mass approximately equal to 5.97 x 10^24 kg, which is very close to NASA's Earth Fact Sheet value of 5.9726 x 10^24 kg.