Question

An astronaut wishes to measure the atmospheric pressure on Mars using a mercury barometer like that shown in the figure, The calibration of the barometer is the standard calibration for Earth: 760 mmHg corresponds to the pressure due to Earth's atmosphere, 1 atm or \(101.325 \mathrm{kPa}\). How does the barometer need to be recalibrated for use in the atmosphere of Mars -by what factor does the barometer's scale need to be "stretched"? In her handy table of planetary masses and radii, the astronaut finds that Mars has an average radius of \(3.39+10^{5} \mathrm{~m}\) and a mass of \(6.42 \cdot 10^{23} \mathrm{~kg}\).

Short answer

Step 1: Calculate gravitational acceleration on Mars Using the formula and values given: \(g_{Mars} = G \frac{M_{Mars}}{R_{Mars}^2} = 6.674 \cdot 10^{-11} \frac{6.42 \cdot 10^{23}}{(3.39 \cdot 10^6)^2}\) \(g_{Mars} ≈ 3.71 \mathrm{~m/s^2}\) Step 2: Calculate the atmospheric pressure on Mars Using the pressure ratio and values: \(P_{Mars} = P_{Earth} \cdot \frac{g_{Mars}}{g_{Earth}} = 101.325 \frac{3.71}{9.81}\) \(P_{Mars} ≈ 38.56 \mathrm{kPa}\) Step 3: Determine the recalibration factor Using the pressure values calculated: \(\text{Recalibration factor} = \frac{P_{Mars}}{P_{Earth}} = \frac{38.56}{101.325}\) \(\text{Recalibration factor} ≈ 0.38\) Step 4: Apply the recalibration factor to the barometer scale Applying the recalibration factor to the Earth's barometer calibration (760 mmHg): \(760 \mathrm{mmHg} \cdot 0.38 ≈ 288.8 \mathrm{mmHg}\) Therefore, the barometer scale needs to be stretched by a factor of 0.38 for use on Mars. The new calibration on Mars is approximately 288.8 mmHg.

Step-by-step solution

Calculate gravitational acceleration on Mars

To find the gravitational acceleration on Mars, we can use the formula: \(g_{Mars} = G \frac{M_{Mars}}{R_{Mars}^2}\) where \(G = 6.674 \cdot 10^{-11} m^3·kg^{–1}·s^{–2}\) is the gravitational constant, \(M_{Mars} = 6.42 \cdot 10^{23} \mathrm{~kg}\) is the mass of Mars, \(R_{Mars} = 3.39 \cdot 10^{6} \mathrm{~m}\) is the radius of Mars.

Calculate the atmospheric pressure on Mars

To calculate the atmospheric pressure on Mars, we can use Earth's atmospheric pressure as a reference. The pressure on Mars can be found using the ratio of the gravitational accelerations: \(P_{Mars} = P_{Earth} \cdot \frac{g_{Mars}}{g_{Earth}}\) where \(P_{Earth} = 101.325 \mathrm{kPa}\), \(g_{Earth} = 9.81 \mathrm{~m/s^2}\).

Determine the recalibration factor

To find the recalibration factor, we can simply divide the atmospheric pressure on Mars by Earth's atmospheric pressure: \(\text{Recalibration factor} = \frac{P_{Mars}}{P_{Earth}}\)

Apply the recalibration factor to the barometer scale

To recalibrate the barometer for use on Mars, we need to multiply the original Earth calibration (760 mmHg) by the recalibration factor calculated in the previous step. This will give us the new calibration, in mmHg, that corresponds to the atmospheric pressure on Mars. Now, let's calculate the values for each step.