Tiêu đề 5. GRAVITATION WS-3 (108-118).pmd.pdf ✅

Olympiad Class Work Book VIII – Physics (Vol – III) Gravitational potential: Gravitational potential at a point in a gravitational field is defined as the amount of work done in bringing a unit mass from infinity to that point. Let “w” be the work done in bringing the test mass m0 from infinity to a point P in a gravitational field, then gravitational potential at point P is given by 0 W V= m SI unit of gravitational potential is J/kg. Dimensional formula of gravitational potential is 0 2 -2   M L T   . Gravitational potential due to point mass: Gravitational potential at a distance ‘r’ from the point mass having mass M is given by G M V=- r O M P r dr  The variation of gravitational potential due to a point mass with distance from the centre is given below. r V (r) –GM r Gravitational potential due to circular ring: Gravitational potential due to a circular ring, at a distance ‘x’ from the center and on the axis of a ring of mass M and radius ‘R’ is given by
VIII – Physics (Vol – III) Olympiad Class Work Book dm 2 2 x R x R P 2 2 -GM V= R +x At the center of ring i.e., -GM 0; V= R x  The variation of gravitational potential due to a circular ringwith distance from the centre is given below. X V (X) –GM –GM 2 2 R X R Gravitational potential due to spherical shell: Let M be the mass of spherical shell and R be it’s radius. The gravitational poten- tial at a distance ‘r’ from the centre of the spherical shell is given by -GM V= r R r P1 P2 P3 Special cases: 1. At any point P1 inside the spherical shell if r < R inside -GM V = R 2. At any point P2 on the surface of the spherical shell if r = R surface -GM V = R 3. At a point P3 outside the spherical shell if r > R
Olympiad Class Work Book VIII – Physics (Vol – III) outside -GM v = r 4. At infinity , 0 v  The variation of gravitational potential due to a circular ring with the distance from the centre is given below. O x R R V GM R r Gravitational potential due to solid sphere: Let M be the mass of solid sphere and R be it’s radius. The gravitational potential at a distance ‘r’ from the centre of the sphere is as given below. R r P1 P2 P3 special cases: 1. At any point P1 inside the Solid sphere if r < R   2 2 inside 3 GM V =- 3R -r 2R 2. At any point P2 on the surface of the solid sphere if r = R surface -GM V = R 3. At a point P3 outside the Solid sphere if r > R outside -GM v = r 4. Gravitational potential due to solid sphere at the centre centre 3 GM V =- 2 R
VIII – Physics (Vol – III) Olympiad Class Work Book i.e., centre surface 3 V = V 2 5. At infinity, 0 v  The variation of gravitational potential due to a circular ring with distance from the centre is given below. O R R V GM R – 3 2 GM R – r Gravitational potential energy (U): The Work done in bringing a particle from infinity to a point in the gravitational field in stored as Gravitational potentialenergy. We know that 0 W V= m W = mV0 U mV  0 i.e., gravitational potential energy = mass of the body x gravitational potential Gravitational potential energy of two particle system: Gravitational potential energy of a system of particles is defined as the external work required to assemble the particles from infinity to a given configuration. P m1 m2 r12 Consider two particles of masses m and m . 1 2 The Gravitational potential due to m1 at P at a distance 1 2 r is 1 12 Gm V = r 