Title 7. GRAVITATION WS-5 (126-131).pmd.pdf ✅

Olympiad Class Work Book VIII – Physics (Vol – III) Kepler’s laws Kepler’s first laws: Planets revolve around the sun in elliptical orbits with the sun at one focus. An ellipse with two foci F1 and F2 is shown in figure. F1 F2 a a b b . The short dimension of the ellipse is called minor axis having length 2b. The long dimension of the ellipse is called major axis having length 2a. The closest point to the sun, in the orbit is called perihelion. The farthest point to the sun, in the orbit is called aphelion. Kepler’s second laws: The radius vector drawn from the sun to the planet sweeps out equal areas in equal intervals of time. equal area in equal intervals of tiem Planet i.e., the areal velocity of the planet around the sun is constant. Areal velocity = d A d t = constant
VIII – Physics (Vol – III) Olympiad Class Work Book Let the planet moves from point A to B in small time interval ‘dt’ with orbital velocity v0 . The radius vector sweeps an area dθ , in this time interval. The area swept by the radius vector 1 2 dA= r dθ 2 Areal velocity dA 1 2 = dt 2 d r dt dA 1 2 = r ω dt 2 2 0 dA 1 v = r dt 2 r       Since v =r 0 ω 0 d A 1 = rv d t 2 d A L = d t 2 m Since L =mv r0 Note: Kepler’s second law is the consequence of Law of conservation of angular momen- tum. Kepler’s third law: The square of time period of a planet is proportional to the cube of the semi major axis of the elliptical orbit i.e., 2 3 T α a Where ‘a’ is the length of the semi major axis. (or) The square of the time period of a planet is directly proportional to the cube of the radius of the orbit, if the orbit is assumed to be a circle. i.e., 2 3 T α r Proof: Let us consider a planet of mass ‘m’ revolving around another planet of mass M, in circular orbit of radius ‘r’. The necessary centripetal force required is provided by gravitational force of attraction. If ‘v’ be the velocity of the revolving planet, then 2 2 mv Mm =G r r   2 GM v = 1 r     
Olympiad Class Work Book VIII – Physics (Vol – III) 2πr but, T= v   2πr v= 2 T      from (1) & (2) 2 2πr GM T r        2 2 3 4π T = r GM       2 3 T α r CUQ 1. According to kepler’s second law, line joining the planet to the sun sweeps out equal areas in equal time intervals. This suggest that for the planet. 1) radial acceleration is zero 2) tangential acceleration is zero 3) transverse acceleration is zero 4) all 2. According to kepler’s first law 1) planets revolve in circular orbits in regular intervals 2) planets revolve in circular orbits in irregular intervals 3) planets revolve in elliptical orbits in regular intervals 4) planets revolve in elliptical orbits in irregular intervals 3. According to kepler’s third law 1) 2 3 T α a 2) 2 T α a 3) T α a 4) 3 T α a 4. Areal velocity of a planet dA dt  1) L m 2) L 2m 3) m L 4) 2L m 5. If ‘a’ is areal velocity of a planet of mass m, its angular momentum is 1) M/A 2) 2MA 3) A2M 4) AM2
VIII – Physics (Vol – III) Olympiad Class Work Book Single Answer Choice Type: 1. The period of revolution of a surface satellite around a planet of radius R is ‘T’. The period of revolution around another planet whose radius is 3R is 1) T 2) 3T 3) 4T 4) 3 3 T 2. The ratio of the distance of two planets from the sun is 1:2. Then ratio of their periods of revolutions is 1) 1 : 4 2) 1: 2 3) 1:2 4) 1:2 2 3. The distance of Neptune and Saturn from sun are nearly 1013 and 1012 meters respectively. Assuming that they move in circular orbits, their periodic times will be in the ratio 1) 10 2) 100 3) 10 10 4) 1 10 4. If the earth is at one-fourth of its present value from the sun, the duration of the day will be 1) half the present year 2) one - eighth the present year 3) one - fourth the present year 4) one - sixth the present year 5. A geostationary satellite is take into an orbit such that its radius is twice as its initial radius. Then the period of its revolution in the second orbit is 1) 48 hrs 2) 24 2hrs 3) 48 2hrs 4) 24 hrs 6. Imagine a light planet revolving around a massive star in circular orbit of radius ‘R’ with a period of revolution T. if the gravitational force of attraction between the planet and the star is proportional to 5 R2 then 2 T is proportional to 1) 3 R 2) 7 R2 3) 5 R2 4) 3 R 2 7. A planet moves around the sun. At a given point P, It is closet from at a distance d1 and has velocity v1 . At another point Q, when it is farthest from the sun at a distance d2 , its speed will be 1) 1 1 2 d v v 2) 2 1 1 2 2 d v d 3) 2 1 1 d v d 4) 2 2 1 2 1 d v d 8. Two planets at mean distance d1 and d2 from the sun, and their frequencies are n1 and n2 respectively. Then 1) 2 2 2 2 n d =n d 1 1 2 2 2) 2 3 2 3 n d =n d 2 2 1 1 3) 2 2 n d =n d 1 1 2 2 4) 2 2 n d =n d 1 1 2 2