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Physics Lesson 4.2.3 - What Quantity Characterizes Mathematically the Gravitational Field?
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Welcome to our Physics lesson on What Quantity Characterizes Mathematically the Gravitational Field?, this is the third lesson of our suite of physics lessons covering the topic of Types of Forces I. Gravitational Force and Weight, you can find links to the other lessons within this tutorial and access additional physics learning resources below this lesson.
What Quantity Characterizes Mathematically the Gravitational Field?
In the previous example, we saw that a 1 kg object is attracted by 9.81 N gravitational force. From the equation (1.b) it is obvious that if the mass of the object was 2 kg, the gravitational force produced by the Earth would be 2 × 9.81N = 19.62 N, for a 3 kg object the gravitational force is 3 × 9.81 N = 29. 43 N and so on. Therefore, the relationship between the object's mass m and the resulting gravitational force Fg produced by the Earth near its surface is
Equation 2
F⃗g = m × 9.81The factor 9.81 is not new. In the Kinematics chapter we stated that the value of gravitational acceleration near the Earth surface is g⃗ = 9.81 m/s2. If using the dimensional analysis discussed in the Physics tutorial "Length, Mass and Time. Dimensional Analysis", we can find that the coefficient 9.81 obtained in the equation (2) has the following dimensions:
=>the unit of the constant 9.81 is [N/kg]
= [kg × m/s2 × kg]
= [m/s2]
(By definition, 1 N is equal to 1kg × m/s2)
Therefore, we obtained the same unit as that of acceleration. Hence, we can conclude that the constant 9.81 we found earlier, is nothing more but the gravitational acceleration (the acceleration of free fall discussed in Kinematics). Thus, the equation (2) for the gravitational force becomes
Equation 3
F⃗g = m × g⃗In Dynamics, i.e. in the part of Physics dealing with forces, the gravitational acceleration g⃗ is otherwise known as the "gravitational field strength." Therefore, depending on the context, g⃗ can be expressed in different names.
To summarize, gravitational field strength (gravitational acceleration) g⃗ is the physical quantity, which characterizes mathematically the gravitational field. Its value is g⃗ = 9.81 m/s2 = 9.81 N/kg near the Earth surface but it becomes smaller when moving away from the Earth. It can be calculated by the equation
Equation 4
g⃗ = F⃗g/mwhere m is the mass of the object, (not the Earth). Substituting the equation (1.b) in the numerator instead of F⃗g, we obtain
Equation 5
g⃗ = G × M × m/R2/m = >g⃗ = G × M/R2
From the equation (5) it is obvious that the gravitational field strength g⃗ generated by the Earth, does not depend on the properties of the other object which is inside this gravitational field, but only on the distance R from the centre of the Earth (the other quantities such as mass of the Earth M and the gravitational constant G are constants).
Example 2
What is the magnitude of gravitational field strength g⃗ at 3629 km above the Earth surface? Take Mass of the Earth equal to 5.972 × 1024 kg and Radius of the Earth equal to 6371 km.
Solution 2
In this example, we have to calculate the height h from the ground as shown in the figure.
We must add h to the Earth's Radius R in the equation (5), to express the distance between the objects. Thus, giving that h = 3629km = 3 629 000m = 3.629 × 106m, we have
Equation 6
g⃗ = G × M/(R + h)2= 6.674 × 10-11 × 5.972×1024/(6.371×106 + 3.629 × 106 )2
= 6.674 × 10-11 × 5.972 × 1024/(10 × 106)2
= 6.674 × 10-11 × 5.972 × 1024/( 107 )2
= 39.857 × 1013/1014
= 39.857 × 10-1
= 3.9857 m/s2
≈ 4m/s2
This result means that at h = 3629 km above the Earth's surface, the attraction of gravity is only 40% of its value on the ground. This is the main reason why there is no air at that height, i.e. the Earth is not able to hold the air at such heights and as a result, the air escapes in the space. (More than 99% of the air is in the first 50 km of the atmosphere).
You have reached the end of Physics lesson 4.2.3 What Quantity Characterizes Mathematically the Gravitational Field?. There are 5 lessons in this physics tutorial covering Types of Forces I. Gravitational Force and Weight, you can access all the lessons from this tutorial below.
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