How Accurate Is the Calculation of Potential Energy Change at High Altitudes?

[studentxlol]

Homework Statement

Calculate the gain of potential energy of a mountaineer of mass 80kg who travels to the top of the mountain Everest 9km above sea level.

Homework Equations

ΔEp=mgΔh

The Attempt at a Solution

ΔEp=mgΔh=80x9.81x9x10^3=7.1x10^6J.

This is the correct answer but surely it's only an estimate since the value of g 9km above the Earth's surface is 9.83830. Δg=9.8380-9.81= 2.8x10^(-3)?

Is it possible to calculate the exact change is Ep as the mass moves through Δh, rather than using a fixed value of 9.81?

 

[Hootenanny]

Homework Statement

Calculate the gain of potential energy of a mountaineer of mass 80kg who travels to the top of the mountain Everest 9km above sea level.

Homework Equations

ΔEp=mgΔh

The Attempt at a Solution

ΔEp=mgΔh=80x9.81x9x10^3=7.1x10^6J.

This is the correct answer but surely it's only an estimate since the value of g 9km above the Earth's surface is 9.83830. Δg=9.8380-9.81= 2.8x10^(-3)?

Is it possible to calculate the exact change is Ep as the mass moves through Δh, rather than using a fixed value of 9.81?

Yes. One would need to go back to Newton's law of gravitation and compute

$$\Delta U = \int_{R}^{R+h} \frac{GM_E m}{r^2}\;\text{d}r,$$

where R is the radius of the Earth, M_E is the mass of the Earth, m is the mass of the man and h is the height of Everest.

 

[studentxlol]

Yes. One would need to go back to Newton's law of gravitation and compute

$$\Delta U = \int_{R}^{R+h} \frac{GM_E m}{r^2}\;\text{d}r,$$

where R is the radius of the Earth, M_E is the mass of the Earth, m is the mass of the man and h is the height of Everest.

Ah ok.

How would you go about integrating GMm/r^2? My maths isn't great but this is what I get:

Since G is a constant, factor it out.

$$\Delta U = G\int_{R}^{R+h} \frac{M_E m}{r^2}\;\text{d}r,$$

$$\Delta U = G [-\frac{M_E m}{r}]_{R}^{R+h}$$

Sorry, my maths isn't at this level yet!

 

[Hootenanny]

Ah ok.

How would you go about integrating GMm/r^2? My maths isn't great but this is what I get:

Since G is a constant, factor it out.

$$\Delta U = G\int_{R}^{R+h} \frac{M_E m}{r^2}\;\text{d}r,$$

$$\Delta U = G [-\frac{M_E m}{r}]_{R}^{R+h}$$

Sorry, my maths isn't at this level yet!

So we are left with
$$\Delta U = GM_E m \left(\frac{1}{R} - \frac{1}{R+h}\right).$$
Simply plug the numbers in and you should get your result.

 

[rwisz]

I would like to clarify that the value of gravitational acceleration (g) does not change significantly at different altitudes on Earth. While it may vary slightly due to factors such as the Earth's rotation and density, the difference is negligible for practical calculations such as this one. Therefore, using a fixed value of 9.81 m/s^2 is a valid and accurate approach for calculating the change in potential energy.

However, if you would like to calculate the exact change in potential energy as the mass moves through Δh, you can use the equation ΔEp = -GmM(1/rf - 1/ri), where G is the universal gravitational constant, m is the mass of the object, M is the mass of the Earth, rf is the final distance from the center of the Earth, and ri is the initial distance from the center of the Earth. This equation takes into account the varying gravitational field strength at different distances from the Earth's center.

In the case of the mountaineer traveling to the top of Mount Everest, you can use the average radius of the Earth (6,371 km) as the initial distance (ri) and the sum of the Earth's radius and the altitude of Mount Everest (6,380 km) as the final distance (rf). This will give you a more precise calculation of the change in potential energy. However, the difference from the estimation using a fixed value of 9.81 will still be very small.