How Much Work is Required to Lift a Payload on the Moon?

In summary, the question involves finding the work done in lifting a 4000kg payload from the surface of the moon to a height of 25000m above its surface. The formula for work done is W = \int F(r).dr, where F(r) is the force and dr is the distance. The gravitational force between two masses m1 and m2 at a distance d is given by F=GMM/d^2. The work done can be found by substituting the given values and calculating the integral. The correct value is slightly smaller due to the non-constant nature of the gravitational force.
  • #1
smileandbehappy
66
0
Ok I have an answer for this but my method is so simple is must be wrong.

Question

The work done against a firce F(r) in moving an object from r=r1 to r=r2 is integralF(r) dr limits R2 at top and r1 at bottom. The gravitational attraction between two masses m1 and m2 at distance d is given be F=GMM/d^2, G=6.67*10^-11. Find the work done in lifting a 4000kg payload from the surface on the moon to a height of 25000m above its surface. The mass of the moon can be taken as 7.3*10^22 and its radius at 1.7*10^6.


My answer.

I subbed in the values for F getting 6739.24 then integrated with respect to r within the limits of 25000 and 0 and got an answer of 1.68*10^8j! What am i doing wrong?
 
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  • #2
Yep. You have made a mistake. Work done is calculated using line integrals.
[tex] W = \int F(r).dr = \int \frac{GM_1M_2}{r^2} dr = -\frac{GM_1M_2}{r} + C[/tex]

Since the integral is definite, then the constant of integration can be ignored, and the answer can be found simply by putting the values given into the equation.

[tex] W = \frac{GM_1M_2}{r_1} - \frac{GM_1M_2}{r_2} [/tex]
 
  • #3
I gather the OP was questioning the magnitude of the answer. Since the gravitational force is approximately constant over the 25 km range in question, the value 1.68e8 joules is roughly correct. The correct value (see post #2) is slightly smaller because the gravitational force is not constant. It takes a lot of energy to lift something off the Moon.
 

What is work done integration?

Work done integration is a mathematical concept used to calculate the total amount of work done by a force over a given distance. It involves finding the area under a force-distance curve using integration techniques.

What is the basic formula for work done integration?

The basic formula for work done integration is W = ∫F(x)dx, where W represents the work done, F(x) is the force as a function of position, and dx is an infinitesimal change in position.

What are some common units used for work done integration?

The most common units used for work done integration are joules (J) and newton-meters (Nm). However, depending on the specific situation, other units such as calories or foot-pounds may also be used.

How is work done integration related to kinetic energy?

Work done integration is related to kinetic energy through the work-energy theorem, which states that the work done on an object is equal to the change in kinetic energy of that object. This means that the area under the force-distance curve represents the change in kinetic energy of the object.

What are some real-life applications of work done integration?

Work done integration is used in various fields such as physics, engineering, and mechanics to calculate the work done by forces in different situations. It can be applied to problems involving lifting objects, pushing or pulling objects, and the motion of objects on inclined planes, among others.

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