9. The surface of the earth is approximately 6,400 km from its center. If the mass of the earth is 6.0 x 1024 kg, what is the acceleration due to gravity near the surface?
The acceleration due to gravity derives from g = µ/R^2 where
G = the acceleration due to gravity
µ = the earths gravitational constant = GM
G = the universal gravitational constant = 6.67259x10^-11
M = the mass of the earth = 5.97424x10^24
R = the mean surface radius of the earth = 6378km
Therefore, g = 6.67259x10-11(5.97424x10^24)/6,378,000^2
……………..= 9.7995m/sec^2
To calculate the acceleration due to gravity near the surface of the earth, we can use Newton's law of universal gravitation. The formula is:
g = G * M / r^2
Where:
g = acceleration due to gravity
G = gravitational constant (approximately 6.67 x 10^-11 N.m^2/kg^2)
M = mass of the earth (6.0 x 10^24 kg)
r = distance from the center of the earth to the surface (6,400 km)
First, let's convert the distance from km to meters:
r = 6,400 km * 1,000 m/km = 6,400,000 m
Now, we can substitute the values into the formula:
g = (6.67 x 10^-11 N.m^2/kg^2) * (6.0 x 10^24 kg) / (6,400,000 m)^2
Simplifying the equation:
g = (6.67 x 6.0 x 10^13) / (6,400,000)^2
g ≈ 9.8 m/s^2
Therefore, the acceleration due to gravity near the surface of the earth is approximately 9.8 m/s^2.
To find the acceleration due to gravity near the surface of the earth, we can use Newton's law of universal gravitation. The formula is given by:
F = (G * M * m) / r^2
Where:
F = force of gravity
G = gravitational constant (approximately 6.67 x 10^-11 Nm^2/kg^2)
M = mass of the earth (6.0 x 10^24 kg)
m = mass of the object (we can assume 1 kg for simplicity)
r = distance from the center of the earth to the object (6,400 km or 6,400,000 meters)
To find the acceleration due to gravity, we need to solve for F in the equation F = ma (Newton's second law of motion), where a is the acceleration and m is the mass. Therefore, we can rearrange the equation as:
F = m * a
(G * M * m) / r^2 = m * a
Next, we can cancel out the mass (m) on both sides of the equation:
(G * M) / r^2 = a
Now, we can plug in the values:
(G * M) / r^2 = a
(6.67 x 10^-11 Nm^2/kg^2 * 6.0 x 10^24 kg) / (6,400,000 m)^2 = a
Solving this equation will give you the value of the acceleration due to gravity near the surface of the earth.