A sensitive gravimeter at a mountain observatory finds that the free-fall acceleration is 0.0070m/s2 less than that at sea level (gsealevel = 9.83 m/s2). What is the observatory's altitude? Assume Rearth = 6.37 × 106 m.
To find the altitude of the observatory, we need to use the equation for acceleration due to gravity:
g = (G * M) / r^2
Where:
- g is the acceleration due to gravity at a given altitude
- G is the gravitational constant (approximately 6.67430 × 10^-11 m^3⋅kg^−1⋅s^−2)
- M is the mass of the Earth
- r is the distance from the center of the Earth to the altitude of the observatory
At sea level, the free-fall acceleration is given as gsealevel = 9.83 m/s^2.
We can rearrange the equation to solve for r:
r^2 = (G * M) / g
Let's substitute the known values into the equation:
r^2 = (6.67430 × 10^-11 m^3⋅kg^−1⋅s^−2 * M) / 9.83 m/s^2
Now we need to find the value of M, which is the mass of the Earth. The mass of the Earth is approximately 5.972 × 10^24 kg.
r^2 = (6.67430 × 10^-11 m^3⋅kg^−1⋅s^−2 * 5.972 × 10^24 kg) / 9.83 m/s^2
Now, let's solve for r:
r = √((6.67430 × 10^-11 m^3⋅kg^−1⋅s^−2 * 5.972 × 10^24 kg) / 9.83 m/s^2)
Note that sqrt represents square root.
Once you have the value of r, subtract the radius of the Earth (6.37 × 10^6 m) to find the altitude of the observatory:
Altitude = r - 6.37 × 10^6 m
Calculate the expression, and you will get the altitude of the observatory.