Lesson 2 of 11 · 6 min
The law of areas
NCERT §7.2
Through the year, the sun's disc seen through Kavya's filter changes size very slightly: it is biggest in early January, when the earth is closest. At that time the earth also moves fastest along its orbit. The two facts are tied by one law.
The lesson in notes
In short
Law of areas: in any two equal stretches of time, the sun–planet line covers the same area. A planet therefore speeds up near the sun and slows down far from it.
In a short time Δt the sun–planet line sweeps ΔA = ½ |r × v Δt|, so ΔA/Δt = L/(2m), where L = r × p is the planet's angular momentum.
Gravity on a planet points along the sun–planet line (a central force), so it exerts no torque about the sun; L stays constant and so does ΔA/Δt.
The law of areas therefore holds for any central force, not only for an inverse-square one.
At perihelion and aphelion r and v are perpendicular, so m r_P v_P = m r_A v_A, giving v_P / v_A = r_A / r_P: the planet is fastest at perihelion.
The half of the orbit around the aphelion encloses the larger area, so the planet spends the longer time on that half.
Watch a class
Prefer a video? Watch this
Equal areas and angular momentum
Khan Academy India - English · English · Lecture · Open on YouTube