# Kepler's second Law

Figure 13.19 shows Kepler’s second law. In a small time interval dt, the line from the sun S to the planet P turns through an angle du. The area swept out is the colored triangle with height r, base length r du, and area dA = 1/ 2 r2 du in Fig. 13.19b. The rate at which area is swept out, dA/dt, is called the sector velocity:

The essence of Kepler’s second law is that the sector velocity has the same value at all points in the orbit. When the planet is close to the sun, r is small and du/dt is large; when the planet is far from the sun, r is large and du/dt is small.

To see how Kepler’s second law follows from Newton’s laws, we express dA/dt in terms of the velocity vector v S of the planet P. The component of v S perpendicular to the radial line is v. = vsinØ. From Fig. 13.19b the displacement along the direction of v. during time dt is r du, so we also have v# = r du/dt. Using this relationship in Eq. (13.14), we find

Now rvsinØ is the magnitude of the vector product r X S : v X S , which in turn is 1/m times the angular momentum L X S = r X S : mv S of the planet with respect to the sun. So we have

Thus Kepler’s second law—that sector velocity is constant—means that angular momentum is constant!

It is easy to see why the angular momentum of the planet must be constant. According to Eq. (10.26), the rate of change of L S equals the torque of the gravitational force F S acting on the planet:

In our situation, r S is the vector from the sun to the planet, and the force F → is directed from the planet to the sun (Fig. 13.20). So these vectors always lie along the same line, and their vector product r S : F S is zero. Hence dL → /dt = 0. This conclusion does not depend on the 1/r2 behavior of the force; angular momentum is conserved for any force that acts always along the line joining the particle to a fixed point. Such a force is called a central force. (Kepler’s first and third laws are valid for a 1/r2 force only.)

Conservation of angular momentum also explains why the orbit lies in a plane. The vector L S = r S : mv S is always perpendicular to the plane of the vectors r S and v S ; since L S is constant in magnitude and direction, r → and v → always lie in the same plane, which is just the plane of the planet’s orbit.

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Ans: First consider the elliptical orbits described in Kepler’s first law. Figure 13.18 shows the geometry of an ellipse. The longest dimension is the major axis, with half-length a; this half-length is called the semi-major axis. view more..
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Ans: A circular orbit, like trajectory 4 in Fig. 13.14, is the simplest case. It is also an important case, since many artificial satellites have nearly circular orbits and the orbits of the planets around the sun are also fairly circular view more..
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Ans: In a small time interval dt, the line from the sun S to the planet P turns through an angle du. The area swept out is the colored triangle with height r, base length r du, and area dA = 1 2 r2 du in . The rate at which area is swept out, view more..
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Ans: We have already derived Kepler’s third law for the particular case of circular orbits. Equation (13.12) shows that the period of a satellite or planet in a circular orbit is proportional to the 3 2 power of the orbit radius. view more..
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Ans: In 1916 Albert Einstein presented his general theory of relativity, which included a new concept of the nature of gravitation. In his theory, a massive object actually changes the geometry of the space around it view more..
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Ans: The first expression for escape speed in Eq. (13.29) suggests that a body of mass M will act as a black hole if its radius R is less than or equal to a certain critical radius. view more..
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Ans: At points far from a black hole, its gravitational effects are the same as those of any normal body with the same mass. If the sun collapsed to form a black hole, the orbits of the planets would be unaffected. But things get dramatically different close to the black hole. view more..
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Ans: If light cannot escape from a black hole and if black holes are small . how can we know that such things exist? The answer is that any gas or dust near the black hole tends to be pulled into an accretion disk that swirls around and into the black hole, rather like a whirlpool view more..
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Ans: HERE ISA SUMMARY OF GRAVITATION , FOR QUICK REVISION view more..
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Ans: Many kinds of motion repeat themselves over and over: the vibration of a quartz crystal in a watch, the swinging pendulum of a grandfather clock, the sound vibrations produced by a clarinet or an organ pipe, and the back-and-forth motion of the pistons in a car engine. This kind of motion, called periodic motion or oscillation view more..

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