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.