A sensitive gravimeter at a mountain observatory finds that the free-fall acceleration is 9.00×10−3 {\rm m/s^2} less than that at sea level.What is the observatory's altitude?

To determine the observatory's altitude based on the difference in free-fall acceleration, we can use the following formula:

Δg = g_sea - g_obs

Where:
Δg = Difference in free-fall acceleration
g_sea = Free-fall acceleration at sea level
g_obs = Free-fall acceleration at the observatory's altitude

Given that Δg = 9.00 × 10^(-3) m/s^2, we can plug it into the formula:

9.00 × 10^(-3) m/s^2 = 9.8 m/s^2 - g_obs

Now, we can solve for g_obs:

g_obs = 9.8 m/s^2 - 9.00 × 10^(-3) m/s^2

g_obs = 9.791 m/s^2

To find the observatory's altitude, we need to know the relationship between free-fall acceleration and altitude. The free-fall acceleration decreases with an increase in altitude due to the decrease in the gravitational force.

By using the known relationship:

g_obs = g_sea * (1 - 2h/Re)^2

Where:
h = altitude above sea level
Re = radius of the Earth

We can rewrite the equation to solve for h:

(1 - 2h/Re)^2 = g_obs / g_sea

Since the value of g_sea is 9.8 m/s^2:

(1 - 2h/Re)^2 = 9.791 / 9.8

Now we can solve for h.

To determine the observatory's altitude, we can use the concept of the variation of gravitational acceleration with altitude. The free-fall acceleration at sea level (g_sea) is known to be approximately 9.81 m/s^2, and the sensitive gravimeter at the observatory finds that the free-fall acceleration is 9.00×10^-3 m/s^2 less than that at sea level.

The difference in gravitational acceleration between sea level and the observatory can be attributed to the height difference between sea level and the observatory. Let's denote the observatory's altitude as 'h'.

The general formula for calculating gravitational acceleration with altitude is given by:

g_altitude = g_sea * (1 - (2 * h) / R)

where:
g_altitude is the gravitational acceleration at the observatory's altitude,
g_sea is the gravitational acceleration at sea level (9.81 m/s^2),
h is the observatory's altitude (unknown),
R is the average radius of the Earth (6,371,000 m).

We are given that the difference in gravitational acceleration at the observatory is 9.00×10^-3 m/s^2 less than at sea level. We can set up the equation as follows:

g_altitude = g_sea - 9.00×10^-3

Substituting the expressions for g_altitude and g_sea into the equation:

g_sea * (1 - (2 * h) / R) = g_sea - 9.00×10^-3

Now, we can solve this equation to find the value of 'h', which represents the observatory's altitude.

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