(a) A particle is dropped (from radius a with zero velocity) into the gravitational potential corresponding...

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Physics

(a) A particle is dropped (from radius a with zero velocity)into the gravitational potential corresponding to a statichomogeneous sphere of radius a and density Ï. Calculate how longthe particle takes to reach the other side of the sphere. [Hint:the equation of motion is d²r/dt² =âˆ’GM(r)/r² .]
(b) Calculate the time required for a homogeneous sphere ofradius a and density Ï with no internal pressure support tocollapse to zero radius under its own gravity. [Apply the previousequation of motion to a particle on the surface.]

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4.4 Ratings (1109 Votes)

We have a static homogenous sphereof radius and uniform density A particle is dropped from the surface of the sphere with initial velocity We assume that the particle moves through the diameter of thesphere Now the problem has spherical symmetry as the potentialdepends only on the radial coordinate The gravitational potential energy of the mass at a radial distance from the centre of the sphere is given byHere is the mass of the sphere distributed within an imaginary sphereof radius centred about the origin It is given bywhere is the volume of the imaginary sphere Now since thegravitational force is conservative the force acting on theparticle at a radial distance is given byThe negative signshows that the force is always See Answer

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