The gravitational acceleration, g, on the Earth is proportional to the mass, M, of the Earth and inversely proportional to the square of the distance, r, from the Earth's center. If the gravitational acceleration on the center of the Earth is 9.8 m/s^2, what is the gravitational acceleration at a distance of 6,404,000km from the center of the Earth. The Earth has a radius of 6,378,000m and has a mass of 5.97 x 10^24kg. Give answer to two decimal places.
since F = GMm/r^2
and we are changing only r, then if we change r by a factor of k, F changes by a factor of 1/k^2
6,404,000/6,378,000 = 1.0040765, so
F changes by a factor of 1/1.0081696 = 0.992
so, F = 0.992 * 9.8 = 9.72 m/s^2
Physics
To find the gravitational acceleration at a distance of 6,404,000km from the center of the Earth, let's use the formula for gravitational acceleration:
g = (G * M) / r^2
Where:
g is the gravitational acceleration,
G is the gravitational constant (approximately 6.67430 x 10^-11 N * (m/kg)^2),
M is the mass of the Earth,
r is the distance from the center of the Earth.
Step 1: Convert the given distance from kilometers to meters:
6,404,000 km = 6,404,000,000 meters
Step 2: Substitute the known values into the formula:
g = (6.67430 x 10^-11 N * (m/kg)^2 * 5.97 x 10^24 kg) / (6,404,000,000 m)^2
Simplifying the equation:
g = (G * M) / r^2
= (6.67430 x 10^-11 N * (m/kg)^2 * 5.97 x 10^24) / (6,404,000,000 m)^2
Step 3: Perform the calculations:
g = (6.67430 x 10^-11 N * (m/kg)^2 * 5.97 x 10^24) / (6,404,000,000 m)^2
= (39.825591 x 10^13 N * (m/kg)^2) / (4.1003872 x 10^19 m^2)
Dividing the numerator and denominator by 10^13:
g = (39.825591 N * (m/kg)^2) / (4.1003872 x 10^6 m^2)
Simplifying the equation further:
g ≈ 9.7228 N / m^2
So, the gravitational acceleration at a distance of 6,404,000km from the center of the Earth is approximately 9.72 m/s^2 (rounded to two decimal places).