1. What happens to the gravitational force between two bodies if the mass of one of the bodies is halved and the distance between them is doubled?
2. The gravitational force acts on all bodies. Why, then does an apple fall towards earth but the earth does not move towards the apple?
The formulation of Newton’s law of gravitation is a story of human determination and quest for scientific enquiry. In the first instance, Newton‘s calculations did not work and he put aside his papers, in a drawer, for almost 20 years. It was during the advent of a comet in 1680, and at the prodding of Sir Edmund Halley, his friend, that he again worked on his calculations and, subsequently, obtained excellent results.
For an interesting explanation, visit the link:
http://www.physicsclassroom.com/class/circles/u6l3c.cfm
6.6 The Universal GraviTaTional ConsTanT(G)
The proportionality sign in the Newton’s law of gravitation can be eliminated by putting a constant of proportionality, denoted by G. The equation then becomes
F = G 1 2 2
m m r
here G is called the gravitational constant. It is a universal constant because the gravitational force between two bodies placed at a certain distance remains the same,
wherever these bodies may be placed in the universe. The constant G is also independent of the medium in which the interacting bodies are placed.
The value of G, in SI units, was later on, found to be equal to 6.67 × 10–11 nm2/kg2.
The units of G are obtained by using the fact that the force has to be expressed in newtons.
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The numerical value of G would change with a change in the units of measurement for mass and distance.
The very small value of G points to the fact that the gravitational force is an extremely weak force. In fact, it is the weakest of all the fundamental forces (Can you find out, which are the other fundamental forces?) and it becomes important only when the masses of the bodies involved are very large. That is why the gravitational force plays such an important role in the case of heavenly bodies.
http://www.physicsclassroom.com
ConCept probe
1. If there is a gravitational attractive force between all objects, why do we not feel ourselves attracted towards massive structures in our surroundings?
2. Why is the gravitational force an important force for heavenly (or astronomical) objects?
The value of G was first measured by an English Physicist, Henry Cavendish in the eighteenth century. He achieved this by measuring the small force between lead masses with an extremely sensitive torsion balance. A better method was later developed by Philipp von Jolly.
As long as the sizes of the objects are small compared to the distance between them they can be treated as point objects which simplifies considerably the mathematics of their gravitational interaction. The Sun and Saturn are far enough (in comparison to their sizes) for them to be treated as point particles. If the distance between two objects is very large (in comparison to their sizes), we can take them as point objects. What about the case of bodies on the earth? For such bodies, the earth does not seem like a point object. The answer to this dilemma lies in Newton’s shell theorem:
A uniform spherical shell of matter attracts a body that is outside the shell as if all the mass of the shell were concentrated at its center.
According to this theorem, Earth can be regarded as a ‘point mass’, located at the center of Earth and with mass equal to that of Earth. We usually follow
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this theorem in all our calculations.
5.7 vEctor form of nEwton’s lAw of GrAvItAtIon
Since force is a vector quantity, it must be expressed in a vector form. The gravitational force can also be expressed in a vector form by attaching a unit vector to the expression for this force.
By convention, the direction of unit vector is
always taken as directed from the body experiencing the force (body 1) towards the body exerting the force (body 2). Therefore, F = G 1 2 2 ˆ m m r r If we are calculating the force on body A due to body B, thenˆr will be the unit vector drawn from A towards B. It will be imperative to mention here that this vector notation is consistent with the basic fact that the gravitational force is always an attractive force. This implies that the gravitational force is a central force, and hence the direction of this force has to be along the line joining the center of the two bodies.
5.8 prIncIplE of supErposItIon
Newton‘s law has been stated for two point bodies. How do we calculate the force on a body if there are more than two bodies interacting with one other?
Figure 12: Diagram showing direction of the gravitational force acting along the
line joining the center of two bodies.
1 A 2 B ^ r
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The solution to this situation lies in what is called the principle of superposition. In a group of objects, the net gravitational force, on any one of the objects, is the vector sum of the forces due to all the other objects. The principle implies that we first calculate the gravitational force that acts on an object due to each of the other objects as if all other objects are absent. After doing this for all possible pairs, the net force on the object under consideration is calculated by the vector sum of all the forces acting on it.
→ 1
F = →F12 +→F13+→F14+→F15+...
here →F1 is the net force on object 1 due to all the other objects 2, 3, 4, 5, …
The Principle of superposition is based on the fact that the gravitational interaction between two bodies is independent of the presence of other bodies in the neighborhood.
The same concept is applicable to electrostatic force which will be studied in class XII.
IllustratIon. three objects of masses 5 kg, 3 kg and 3 kg are placed at the corners of an equilateral triangle of side 20 cm. Calculate the net gravitational force on the object of 5 kg.
solutIon. The magnitude of the force on object A due to object B is FAB = Gm mA B2 r It is in the direction of AB Putting the values we get, FAB 5 × 10–9 N
Similarly, the magnitude of the force on A due to body C is
Fac 5 × 10–9 N It is the direction of AC.
As per the principle of superposition, the net force on the body A is the vector sum of forces →FAB and
→ AC
F . Applying the laws of vector addition, A
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we get the magnitude of the resultant as Fa = 2 + 2 +2 cos60° AB AC AB AC F F F F Fa = × − + × − + × − 9 2 9 2 9 2 1 2 (5 10 N) (5 10 N) 2(5 10 N) = 5 3 × 10–9 N The resultant force is directed along the bisector of the angle between the two forces →FAB and → AC F .Four equal masses are placed at the corners of a square of side 2cm as shown in the figure. Another mass is placed at the centre of the square. Find the magnitude and direction of the net force on the body kept at the center of the square due to all the other masses.