Page 1 147 Unit - 6 Gravitational Page 2 147 Unit - 6 Gravitational 148 SUMMARY 1. Gravitational force between two point masses is 1 2 2 Gm m F r ? 2. Acceleration due to Gravity (I) on the surface of earth = 2 2 9.81 GM ms R ? ? (II) At a height h from surface of earth 1 2 2 (1 ) 1 ? ? ? ? ? ? ? ? ? ? g h g g R h R if h << R (III) At a depth d form the surfce of earth 1 /(1 ) d g g R ? ? g 1 = g if d = R i. e. on the surface of earth (IV) Effect of rotation of earth at latitude ? g 1 = g â€“ R ? 2 cos 2 ? - at the equator ? = 0 g 1 = g â€“ R ? 2 = minimum value - At the pole ? = 90 0 g 1 = g â€“ R ? 2 = maximum value - At the equator effect of rotation of earth is maximum and value of g is minimum. - At the pole effect of rotation of earth is zero and value of g is maximum. 3. Field Strength ? Gravitational field strength at a point in gravirtational field is defined as, F E ? ? ? m = gravitational force per unit mass ? Due to point mass 2 GM E r ? (towards the mass) 2 r 1 E ? ? Due to solid sphere inside points 3 ? i GM E r R At r = 0, E = 0 at the center At r = R, 2 GM E R ? i.e. on the surface out side points E o = 2 GM or r 2 o r 1 E ? Page 3 147 Unit - 6 Gravitational 148 SUMMARY 1. Gravitational force between two point masses is 1 2 2 Gm m F r ? 2. Acceleration due to Gravity (I) on the surface of earth = 2 2 9.81 GM ms R ? ? (II) At a height h from surface of earth 1 2 2 (1 ) 1 ? ? ? ? ? ? ? ? ? ? g h g g R h R if h << R (III) At a depth d form the surfce of earth 1 /(1 ) d g g R ? ? g 1 = g if d = R i. e. on the surface of earth (IV) Effect of rotation of earth at latitude ? g 1 = g â€“ R ? 2 cos 2 ? - at the equator ? = 0 g 1 = g â€“ R ? 2 = minimum value - At the pole ? = 90 0 g 1 = g â€“ R ? 2 = maximum value - At the equator effect of rotation of earth is maximum and value of g is minimum. - At the pole effect of rotation of earth is zero and value of g is maximum. 3. Field Strength ? Gravitational field strength at a point in gravirtational field is defined as, F E ? ? ? m = gravitational force per unit mass ? Due to point mass 2 GM E r ? (towards the mass) 2 r 1 E ? ? Due to solid sphere inside points 3 ? i GM E r R At r = 0, E = 0 at the center At r = R, 2 GM E R ? i.e. on the surface out side points E o = 2 GM or r 2 o r 1 E ? 149 At r ? ? , E ? 0 ? Due to a sphericell shell - inside points E=0 outside points E 0 = 2 GM r just outside surface 2 GM E R ? on the surface E â€“ r graph is discontinuous ? on the axis of a ring 3 2 2 2 ( ) ? ? r GMr E R r At r = 0, E = 0 i.e. at the center If r >> R, 2 ? GM E r , i.e. ring behaves as a points mass As 0 ? ? ? r E 4. Gravitational potential :- (i) Gravitational potential at a point in a gravitational field is defined as the negative of work done in moving a unit mass from infinity to that point per unit mass, thus p p w w V m ? ? ? ? (ii) Due to point mass Gm V rm ? ? 0 0 ? ? ? ? ? ? v as r and v as r (iii) Due to solid sphere ? inside points 2 2 â€“ (1.5 0.5 ) GM Vi R r R ? ? as r = R ? ? GM V R i.e. on the surface V - r graph is parabola for inside points ? out side points GM v r ? ? (iv) Due to sphercal shell inside points GM Vi R ? ? outside points ? ? GM Vi R Page 4 147 Unit - 6 Gravitational 148 SUMMARY 1. Gravitational force between two point masses is 1 2 2 Gm m F r ? 2. Acceleration due to Gravity (I) on the surface of earth = 2 2 9.81 GM ms R ? ? (II) At a height h from surface of earth 1 2 2 (1 ) 1 ? ? ? ? ? ? ? ? ? ? g h g g R h R if h << R (III) At a depth d form the surfce of earth 1 /(1 ) d g g R ? ? g 1 = g if d = R i. e. on the surface of earth (IV) Effect of rotation of earth at latitude ? g 1 = g â€“ R ? 2 cos 2 ? - at the equator ? = 0 g 1 = g â€“ R ? 2 = minimum value - At the pole ? = 90 0 g 1 = g â€“ R ? 2 = maximum value - At the equator effect of rotation of earth is maximum and value of g is minimum. - At the pole effect of rotation of earth is zero and value of g is maximum. 3. Field Strength ? Gravitational field strength at a point in gravirtational field is defined as, F E ? ? ? m = gravitational force per unit mass ? Due to point mass 2 GM E r ? (towards the mass) 2 r 1 E ? ? Due to solid sphere inside points 3 ? i GM E r R At r = 0, E = 0 at the center At r = R, 2 GM E R ? i.e. on the surface out side points E o = 2 GM or r 2 o r 1 E ? 149 At r ? ? , E ? 0 ? Due to a sphericell shell - inside points E=0 outside points E 0 = 2 GM r just outside surface 2 GM E R ? on the surface E â€“ r graph is discontinuous ? on the axis of a ring 3 2 2 2 ( ) ? ? r GMr E R r At r = 0, E = 0 i.e. at the center If r >> R, 2 ? GM E r , i.e. ring behaves as a points mass As 0 ? ? ? r E 4. Gravitational potential :- (i) Gravitational potential at a point in a gravitational field is defined as the negative of work done in moving a unit mass from infinity to that point per unit mass, thus p p w w V m ? ? ? ? (ii) Due to point mass Gm V rm ? ? 0 0 ? ? ? ? ? ? v as r and v as r (iii) Due to solid sphere ? inside points 2 2 â€“ (1.5 0.5 ) GM Vi R r R ? ? as r = R ? ? GM V R i.e. on the surface V - r graph is parabola for inside points ? out side points GM v r ? ? (iv) Due to sphercal shell inside points GM Vi R ? ? outside points ? ? GM Vi R 150 (v) on the axis of a ring 2 2 GM Vr R r ? ? ? at r = 0, ? ? GM V R i.e. at center 5. Gravitational potential Energy (i) This is the work done by gravitational forces in arranging the system from infinite sepration in the present position (ii) Gravitational potential energy of two point massess is 1 2 Gm m U r ? ? (iii) To find the gravitational points energy of more than two points masses we have to make pairs of masses. Neighter of the pair should be repeated. For example in case of four point masses. 4 3 4 2 4 1 3 2 3 1 2 1 43 23 41 32 31 21 m m m m m m m m m m m m U G r r r r r r ? ? ? ? ? ? ? ? ? ? ? ? ? ? for n point masses total number of pairs will be ( 1) 2 n n ? (iv) If a point mass m is plaled on the surface of earth the potential energy here is Uo o GMm U R ? ? and potential energy at height h is h GMm U (R h) ? ? ? the difference in potential energy would be ?U = Uh â€“ Uo = mgh 1 + h/r If h << R, ?U = mgh 6. Relation between field strength ? E and potential V (i) if V is a function of only one variable (Say r) then dV E dr ? ? ? ? slope of U ? r graph (ii) If V is funtion at three coordinates variable; e x, y, and z then v v v Ë† Ë† Ë† E i j k x y z ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Page 5 147 Unit - 6 Gravitational 148 SUMMARY 1. Gravitational force between two point masses is 1 2 2 Gm m F r ? 2. Acceleration due to Gravity (I) on the surface of earth = 2 2 9.81 GM ms R ? ? (II) At a height h from surface of earth 1 2 2 (1 ) 1 ? ? ? ? ? ? ? ? ? ? g h g g R h R if h << R (III) At a depth d form the surfce of earth 1 /(1 ) d g g R ? ? g 1 = g if d = R i. e. on the surface of earth (IV) Effect of rotation of earth at latitude ? g 1 = g â€“ R ? 2 cos 2 ? - at the equator ? = 0 g 1 = g â€“ R ? 2 = minimum value - At the pole ? = 90 0 g 1 = g â€“ R ? 2 = maximum value - At the equator effect of rotation of earth is maximum and value of g is minimum. - At the pole effect of rotation of earth is zero and value of g is maximum. 3. Field Strength ? Gravitational field strength at a point in gravirtational field is defined as, F E ? ? ? m = gravitational force per unit mass ? Due to point mass 2 GM E r ? (towards the mass) 2 r 1 E ? ? Due to solid sphere inside points 3 ? i GM E r R At r = 0, E = 0 at the center At r = R, 2 GM E R ? i.e. on the surface out side points E o = 2 GM or r 2 o r 1 E ? 149 At r ? ? , E ? 0 ? Due to a sphericell shell - inside points E=0 outside points E 0 = 2 GM r just outside surface 2 GM E R ? on the surface E â€“ r graph is discontinuous ? on the axis of a ring 3 2 2 2 ( ) ? ? r GMr E R r At r = 0, E = 0 i.e. at the center If r >> R, 2 ? GM E r , i.e. ring behaves as a points mass As 0 ? ? ? r E 4. Gravitational potential :- (i) Gravitational potential at a point in a gravitational field is defined as the negative of work done in moving a unit mass from infinity to that point per unit mass, thus p p w w V m ? ? ? ? (ii) Due to point mass Gm V rm ? ? 0 0 ? ? ? ? ? ? v as r and v as r (iii) Due to solid sphere ? inside points 2 2 â€“ (1.5 0.5 ) GM Vi R r R ? ? as r = R ? ? GM V R i.e. on the surface V - r graph is parabola for inside points ? out side points GM v r ? ? (iv) Due to sphercal shell inside points GM Vi R ? ? outside points ? ? GM Vi R 150 (v) on the axis of a ring 2 2 GM Vr R r ? ? ? at r = 0, ? ? GM V R i.e. at center 5. Gravitational potential Energy (i) This is the work done by gravitational forces in arranging the system from infinite sepration in the present position (ii) Gravitational potential energy of two point massess is 1 2 Gm m U r ? ? (iii) To find the gravitational points energy of more than two points masses we have to make pairs of masses. Neighter of the pair should be repeated. For example in case of four point masses. 4 3 4 2 4 1 3 2 3 1 2 1 43 23 41 32 31 21 m m m m m m m m m m m m U G r r r r r r ? ? ? ? ? ? ? ? ? ? ? ? ? ? for n point masses total number of pairs will be ( 1) 2 n n ? (iv) If a point mass m is plaled on the surface of earth the potential energy here is Uo o GMm U R ? ? and potential energy at height h is h GMm U (R h) ? ? ? the difference in potential energy would be ?U = Uh â€“ Uo = mgh 1 + h/r If h << R, ?U = mgh 6. Relation between field strength ? E and potential V (i) if V is a function of only one variable (Say r) then dV E dr ? ? ? ? slope of U ? r graph (ii) If V is funtion at three coordinates variable; e x, y, and z then v v v Ë† Ë† Ë† E i j k x y z ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 151 7. Escape velocity (i) From the surface of earth 2 2 2 GM GM Ve gR as g R R ? ? ? =11.2 km / sec (ii) Escape velocity does not depend upon the angle which particle is projected form the surface and the mass of body 8. Motion of satellites (i) orbital speed GM Vo r ? (ii) time period 3/ 2 2 3 2 T r T r GM ? ? ? ? (iii) Kinetic energy 2 ? GMm K r (iv) Potential energy GMm U r ? ? ? ... (v) Total Mechanical energy. GMm E r ? ? 9. Keplerâ€™s laws - First law : Each planet moves in an elliptical orbit with the sun at one focus of ellipse - Second law : The radius vectors drawn form the sun to a planet, sweeps out equal area in equal time interval i.e. areal Velocity is constant. this law is derived from the law of conservation of angular momentum 2 dA L dt m ? = constant here L is the angular momentum and m is mass of planet - Third law 2 3 T r ? where r is semi-major axis of elliptical path - The gravitational force acting between two bodies is always attractive. It is independent of medium between bodies. It holds good over a wide range of distance. It is an action and reaction pair. It is conservative force. It is a central force and obey inverse square law as 2 1/ F r ? - The value of G is never zero any where but the value of g is zero at the center of earth. - the acceleration due to gravity is independent of mass, shape, size etc of falling body. - the rate of decrease of the acceteration due to gravity with height is twice as compared to that with depth. - It the rate of rotation of earth increases the value of acceleration due to gravity decreases at all points on the surface of earth except at poles.Read More

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