How Is Energy Calculated for Moving an Object Away from Earth's Surface?

[aloshi]

hi!
in my book they are trying to derive a formula for how much energy is needed to move an object height h from the Earth's surface. so large that:

dent's total work (W) spent a
to move a body with mass m from the Earth to a point at distance R from the center of the earth:
$$W=c\cdot m\cdot M\cdot (\frac{1}{R_0}-\frac{1}{R})$$
c = 6.66 * 10 ^ -11, R_0 = 6370
when R increases approaching the term 1 / R all zero, and work to keep a body from the Earth's surface infinitely far into the universe can be calculated by the formula
$$W=c\cdot \frac{m\cdot M}{R_0}$$
what I can not really understand is that work is defined as force*distance, W=F*s.
why is $$\frac{1}{R_0}-\frac{1}{R}=distance$$ and why is $$c\cdot m\cdot M=force$$??

can someone explain to me, thanks

2) why is $$c\cdot m\cdot M$$ the same at $$m\cdot g\cdot R^2_0$$, also
$$c\cdot m\cdot M=m\cdot g\cdot R^2_0$$

 

[tiny-tim]

Hi aloshi!

(It's not c, it's G. )

Work isn't force*distance unless the force is constant.

Work is the integral of force wrt distance … W = ∫ F.ds,

and in this case F = GMm/r², so W = ∫ GMm/r² dr = GMm/r + constant.

 

[aloshi]

Hi aloshi!

(It's not c, it's G. )

Work isn't force*distance unless the force is constant.

Work is the integral of force wrt distance … W = ∫ F.ds,

and in this case F = GMm/r², so W = ∫ GMm/r² dr = GMm/r + constant.

but I can not about Integration, can you explain in a different way? pleas

 

[aloshi]

Hi aloshi!

(It's not c, it's G. )

Work isn't force*distance unless the force is constant.

Work is the integral of force wrt distance … W = ∫ F.ds,

and in this case F = GMm/r², so W = ∫ GMm/r² dr = GMm/r + constant.

unless the force is not constant, way they write the worke sow??

and i find this:
$$W=\int_{r=R_0}^{R}Fdr=\int_{r=R_0}^{R}\frac{cmM}{r^2}dr=cmM\int_{r=R_0}^{R}\frac{1}{r^2}dr=cmM\[-\frac{1}{r}\]_{R_0}^R=cmM\(\frac{1}{R_0}-\frac{1}{R}\)$$
but i can not anderstund't

 

[paranormal]

Hello! It looks like you are trying to understand the gravitational formula for work. Let me try to explain it to you.

First, the gravitational force between two objects is given by the equation F = G * m1 * m2 / r^2, where G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between them.

Now, to calculate the work done in moving an object from the Earth's surface to a distance R from the center of the Earth, we use the formula W = F * s, where F is the force and s is the distance moved. In this case, the force is the gravitational force between the object and the Earth, and the distance moved is R - R0, where R0 is the radius of the Earth.

So, we can substitute the formula for F into W = F * s and get W = G * m * M * (1/R0 - 1/R) * s. Here, c is just a constant that combines G, M, and R0, so we can write it as c = G * M / R0. Now, if we plug in this value for c, we get the formula W = c * m * M * (1/R0 - 1/R) * s.

To answer your question about why 1/R0 - 1/R is considered as a distance, it is because it represents the change in distance from the Earth's surface (R0) to a distance R. Similarly, c * m * M represents the force of gravity between the two objects.

As for your second question, c * m * M is not necessarily the same as m * g * R0^2. The latter formula is for the force of gravity at the Earth's surface, where g is the acceleration due to gravity. However, in the formula W = c * m * M * (1/R0 - 1/R) * s, we are considering the work done in moving the object from the Earth's surface to a distance R, which is not necessarily at the Earth's surface. Therefore, the two formulas are not equivalent.

I hope this helps to clarify things for you. If you have any further questions, please let me know. Good luck with your studies!