Central Forces
The rate of change of angular momentum of a single particle about the origin is
.
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The angular momentum of the particle is conserved, If the torque Ï„ =rÃ— F due to the force F on it vanishes.
This happens either when F is zero or when F is along r.
We are interested in forces which satisfy the latter condition. Central forces satisfy this condition.
A 'central' force is always directed towards or away from a fixed point, i.e., along the position vector of the point of application of the force with respect to the fixed point. Further, the magnitude of a central force F depends on r, the distance of the point of application of the force from the fixed point F=F (r).
In the motion under a central force the angular momentum is always conserved.
Two important results follow from this:
- The motion of a particle under the central force is always confined to a plane.
- The position vector of the particle with respect to the centre of the force (i.e., the fixed point) has a constant areal velocity. In other words the position vector sweeps out equal areas in equal times as the particle move sunder the influence of the central force.
It is believed that Newton made calculations in 1666. He did not publish his result for 20 years. Only in 1687 did he publish the law in his book Principia Mathematica. The reason was that he was not satisfied with his conclusions.
In his calculation, the distances are measured from the centre of the earth. This meant that the earth was assumed as a mass point, i.e. the entire mass of the earth was concentrated at its centre for objects outside it.
Newton invented calculus during the intervening years. Using calculus he could prove that, for objects outside the earth, it behaved as though its entire mass was concentrated at the centre. In sec. 8.10 we will prove this without the use of calculus in a special case in which g can be treated as a constant.
Newton went a step further. From his second law of motion (F=ma) he knew that the force exerted on an accelerating body depends on its mass. Thus the force exerted by the earth on the apple (or moon) must depend upon he mass of the apple (or moon).
From his third law motion, the apple (or moon) must exert an equal and opposite force on the earth which must depend upon the mass of the earth. Thus he found that the force of attraction between bodies should depend on their masses and vary inversely as the square of the distance between them.
Newton realized that this attraction is only a particular case of universal attraction between any two bodies situated anywhere in the universe. At the age of twenty-three, he discovered a law, now known a Newton's law of universal gravitation,
which states:
'Any two particles of matter anywhere in the universe attract each other with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them, the direction of the force being along the line joining the particles, i.e.,
where F is the magnitude of the force of attraction between two particles of masses m_{1} and m_{2} separated by a distance r.
Gravitational force |
In the form of an equation the law is written as
Where G is a constant called the universal gravitation constant. The value of this constant is to be determined experimentally and will be the same for any pairs of particles.
The gravitational force is attractive, i.e., the force F is along Ã¢â‚¬â€œr. The force on point mass m_{1} due to m_{2} is of course Ã¢â‚¬â€œF by Newton's third law. Thus, the gravitational force F_{12} on the body 1 due to 2 and F_{21} on the body 2 due to 1 are related as
F_{12}=-F_{21.}
The total force on m_{1 }is given by
Some Features of the law
- The gravitation forces between two particles are an action-and-reaction pair. The first particle exerts a force of magnitude F (on the second particle) that is directed towards the second particle along the line joining the two particles. Likewise, the second particle exerts an equal and opposite force (on the first particle) that is directed towards the first particle along the line joining the two.
- The universal gravitation constant G must not be confused with g which is the acceleration due to gravity. The constant G is scalar having dimension M^{-1} L^{3}T^{-2} and is measured in Nm^{2} kg^{-2}, but g is vector, having dimensions LT^{-2 }and is measured in ms^{-2}, it is neither universal nor constant.
- The constant G is to be determined experimentally. It is universal in the sense that, if G is determined for a given pair of bodies, we can use that value in the law of gravitation to determine the gravitational force between any other pair of bodies.