The radius of the earth is approximately 4,000 miles. The acceleration due to gravity is approximately 32 feet per square second. The mass of the earth is approximately 6 × 10^24 kilograms. Calculate the value of G in units of kilograms, meters, and seconds.
g=F/m=GMe/r^2
G=gr^2/Me
convert g from 32ft/s^2 to m/s^2 then convert rearth from miles to meters.
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To calculate the value of G in units of kilograms, meters, and seconds, we can make use of the equation:
G = (r^2 * g) / M
Where:
G is the gravitational constant,
r is the radius of the Earth,
g is the acceleration due to gravity, and
M is the mass of the Earth.
Let's substitute the given values into the equation to calculate G.
Given:
Radius of the Earth, r = 4000 miles
First, we need to convert the radius of the Earth from miles to meters since G is expressed in kilograms, meters, and seconds.
1 mile is equal to 1609.34 meters.
So, the radius of the Earth in meters would be:
4000 miles * 1609.34 meters/mile = 6437.36 kilometers
Acceleration due to gravity, g = 32 feet per square second
We also need to convert the acceleration due to gravity from feet per square second to meters per square second.
Since 1 foot is equal to 0.3048 meters, and 1 square second is just a second squared, we have:
32 feet/second^2 * 0.3048 meters/foot = 9.7536 meters/second^2
Mass of the Earth, M = 6 x 10^24 kilograms
Now, we can substitute the given values in the equation to calculate G:
G = (r^2 * g) / M
= (6437.36 km)^2 * (9.7536 m/s^2) / (6 x 10^24 kg)
Squaring the radius in meters:
G = (6437.36 km * 1000 m/km)^2 * (9.7536 m/s^2) / (6 x 10^24 kg)
Simplifying the calculation:
G = (6437360 m)^2 * 9.7536 m/s^2 / (6 x 10^24 kg)
= 4.14 x 10^(-3) m^3 / kg / s^2
Therefore, the value of G in units of kilograms, meters, and seconds is approximately 4.14 x 10^(-3) m^3 / kg / s^2.