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Chapter 4: Problem 48

During a balloon ascension, wearing an oxygen mask, you measure the weight of a calibrated \(5.00-\mathrm{kg}\) mass and find that the value of the gravitational field strength at your location is 9.792 N/kg. How high above sea level, where the gravitational field strength was measured to be $9.803 \mathrm{N} / \mathrm{kg},$ are you located?

Short Answer

Expert verified
Answer: The altitude above sea level is approximately 0.796 km or 796 m.

Step by step solution

01

(Step 1: Write down the given information)

: We have the following information: 1. Mass of the object, \(\displaystyle m\ =\ 5.00\ \mathrm{kg}\) 2. The gravitational field strength at altitude, \(\displaystyle g_{1} =9.792\ \mathrm{N} / \mathrm{kg}\) 3. The gravitational field strength at sea level, \(\displaystyle g_{2} =9.803\ \mathrm{N} / \mathrm{kg}\)
02

(Step 2: Find the force acting on the mass)

: We can now find the force acting on the mass \(\displaystyle m\) using the gravitational field strength formula: \(\displaystyle F_{1} = m\times g_{1}\) Plugging in the values, \(\displaystyle F_{1} =5.00\ \mathrm{kg} \times 9.792\ \mathrm{N} / \mathrm{kg} =48.96\ \mathrm{N}\)
03

(Step 3: Use the formula for gravitational field strength at height)

: The formula for calculating the height above sea level, considering that the radius of the Earth \(\displaystyle R\) remains approximately constant, is: \(\displaystyle g_{1} =\dfrac{g_{2} R^{2}}{( R+h)^{2}}\) Where, \(\displaystyle R\) is the Earth's radius (~6371km), \(\displaystyle h\) is the height we want to find, \(\displaystyle g_{1}\) is the gravitational field strength at altitude, and \(\displaystyle g_{2}\) is the gravitational field strength at sea level.
04

(Step 4: Isolate the variable h to solve for height)

: Rearrange the formula to solve for \(\displaystyle h\) by isolating it on one side: \(\displaystyle h=R\left(\dfrac{g_{2}}{g_{1}}-1\right)^{1/ 2}-R\)
05

(Step 5: Calculate the height h)

: Plug in the known values and calculate the height: \(\displaystyle h=6371\ \mathrm{km} \times \left(\dfrac{9.803\ \mathrm{N} / \mathrm{kg}}{9.792\ \mathrm{N} / \mathrm{kg}}\ -\ 1\right)^{1/ 2}\ -\ 6371\ \mathrm{km}\) \(\displaystyle h\approx 0.796\ \mathrm{km}\) Therefore, the height above sea level is approximately \(\displaystyle 0.796\ \mathrm{km}\) or \(\displaystyle 796\ \mathrm{m}\).

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Most popular questions from this chapter

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