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Physics Lesson 8.2.1 - A Quick Recap on Gravitational Potential Energy
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Welcome to our Physics lesson on A Quick Recap on Gravitational Potential Energy, this is the first lesson of our suite of physics lessons covering the topic of Gravitational Potential Energy. Kepler Laws, you can find links to the other lessons within this tutorial and access additional physics learning resources below this lesson.
A Quick Recap on Gravitational Potential Energy
In the Physics tutorial "Gravitational Potential Energy. The Law of Mechanical Energy Conservation", we explained that gravitational potential energy as a special case of potential energy is obtained by multiplying the gravitational force Fg exerted on a system and linear distance r the object moves due to this force, i.e.
in general, and
in the specific case of gravitational potential energy.
Thus, since when multiplying in scalar mode a force and linear distance the object on which this force acts moves, we obtain the work done on this object, we obtain the definition of potential energy.
"Potential energy PE represents the work done by one of the objects in the system (usually the largest) to bring the other object from the position r⃗ to zero, i.e. to bring it at the place where the first object is."
The negative sign provides the convention that work done against a force field increases the potential energy, while work done by the force field decreases the potential energy.
In the specific case of gravitational potential energy, since gravitational force is
where M is the mass of the largest object and m is that of the smallest object, we obtain for the gravitational potential energy possessed by an object when it is at a linear distance R from the Earth
= -G × M × m/r2 × r⃗
= -G × M × m/r
Remark! The simplified equation for gravitational potential energy
can be used only in very specific situations in which the magnitude of gravitational field g is constant, i.e. when objects are very close to Earth surface (for small heights, h). It is known that gravitational field weakens when moving away from the source (here the Earth). Therefore, the only formula we can use to calculate the gravitational field at large distances from the source, is
Example 1
What is the work done by the Earth to move a 4 kg object from h = 29 km above its surface to the ground? Take MEarth = 5.972 × 1024 kg, G = 6.674 × 10-11 N × m2 / kg2 and REarth = 6371 km.
Solution 1
We must calculate the gravitational potential energy at the beginning and the end of process and subtract them in order to calculate the work done by the Earth to send the 4 kg object from h = 29 km to the ground (h = 0).
Clues:
r2 = R = 6371 km = 6 371 000 m = 6.371 × 106 m
MEarth = M = 5.972 × 1024 kg
mobject = m = 4 kg
W = ?
We have:
= -6.674 × 10-11 × 5.972 × 1024 × 4/6.4 × 106
= -2.4910705 × 108 J
and
= -6.674 × 10-11 × 5.972 × 1024 × 4/6.371 × 106
= -2.5024095 × 108 J
Thus, the work done by Earth's gravity on the object is
= -2.4910705 × 108 J - (-2.5024095 × 108 J)
= 0.011339 × 108 J
= 1.1339 × 106 J
Thus, we obtained a result equal to 1 133 900 Joules approximately.
Now, observe what result we would obtain if we used the simplified equation GPE = m × g × h for gravitational potential energy. If we take m = 4 kg, g = 9.81 m/s2 and h = 29 km = 29000 m, the result is
= m × g × h
= 4 × 9.81 × 29000
= 1 137 960 J
The last result is slightly bigger that the result obtained earlier because of the error made when considering the gravity g as constant (9.81 m/s2). It is a known fact that gravity decreases with the increase in altitude. It has a value of 9.81 m/s2 only near the Earth surface. Therefore, in exercises it is better to use as much as possible the general formula of gravitational potential energy
instead of the simplified formula
In the tutorial "Gravitational Potential Energy. The Law of Mechanical Energy Conservation", it was also explained the concept of path independence of gravitational potential energy. This property derives from path independence of gravitational force. This means the path is not important for the values of gravitational force and gravitational potential energy but only the initial and final positions of the object.
You have reached the end of Physics lesson 8.2.1 A Quick Recap on Gravitational Potential Energy. There are 4 lessons in this physics tutorial covering Gravitational Potential Energy. Kepler Laws, you can access all the lessons from this tutorial below.
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