Q.(a) Consider a planet moving in an elliptical orbit around the sun as shown in fig. If t1 and t2 be the time taken by the planet to go from P1 to P2 and from P3 to P4 respectively. The two shaded parts have equal area, then

Kepler's second law (law of areas) says a planet's radius vector from the Sun sweeps equal areas in equal times, which is a direct consequence of conservation of angular momentum.

Statement: The line joining any planet to the Sun (the radius vector) sweeps out equal areas in equal intervals of time as the planet moves along its elliptical orbit — i.e. dA/dt = constant. Because the planet moves faster when nearer the Sun (perihelion) and slower when farther (aphelion), the swept-out area per unit time stays the same. This law follows from the fact th …

Kepler's second law (law of areas): the radius vector from the Sun to a planet sweeps out equal areas in equal time intervals, i.e. dA/dt = constant.

As a planet moves in its elliptical orbit around the Sun, the line (radius vector) joining the Sun to the planet sweeps out area at a constant rate: dA/dt = constant. This means the planet moves faster when it is closer to the Sun (near perihelion) and slower when farther away (near aphelion), so that the swept area per unit time stays the same. This law is a direct consequence of the conservation of angular momentum: since gravity is a central force (always directed along the line joining Sun and planet), it exerts zero torque about …

Kepler's Second Law (the radius vector sweeps equal areas in equal times) follows from conservation of angular momentum, because the Sun's gravitational pull on a planet is a central force with zero torque about the Sun.

The gravitational force the Sun exerts on a planet always points along the line joining the planet to the Sun. A force that always points toward (or away from) a fixed point is called a central force.

Torque about the Sun is tau = r x F. Since F is always parallel (or antiparallel) to r for a central force, the cross product r x F = 0. Zero torque means the angular momentum L = r x p about the Sun does not change with time -- it is conserved.

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