Modern Physics
Modern Physics
3rd Edition
ISBN: 9781111794378
Author: Raymond A. Serway, Clement J. Moses, Curt A. Moyer
Publisher: Cengage Learning
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Chapter 16, Problem 16P

Collapse of a dust ball under Newtonian gravity. Suppose that a spherical ball of dust with no internal pressure and with an initial radius R0 and total mass M gravitationally collapses from rest.

  1. (a) Show that the outer shell of the collapsing dust ball obeys the equation of motion

    R ¨ = G M R 2

  2. (b) Solve this equation of motion and show that the time to collapse is given by

    t c = π 2 R 0 3 2 G M

    Hint: Write Newton’s law as R ¨ = f ( R ) . Let f(R) = dF(R)/dR and multiply both sides of by R ¨ = d F / d R by R ˙ . The resulting equation may be immediately integrated with respect to time by recognizing that R ¨ R ˙ = d d t ( R ˙ 2 2 ) . Integrating again with respect to time, this general technique yields t as a function of R where t = d R 2 ( F ( R ) + C ) + C 1 and C and C1 are constants determined by the initial conditions.

  3. (c) Find the time of collapse of a dust ball with a mass and radius equal to that of the Sun.

    (RS = 6.96 × 1010 cm, MS = 1.99 × 1033 g, G = 6.67 × 10−8 cgs units)

  4. (d) What prevents the Sun from collapsing in this way?
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What is the minimum velocity required for an object of mass “m” to escape the gravitational pull of a planet of mass “M” and radius “R” from its surface? a) [(G*M)/R]1/2 b) [(2*G*M)/R] 2 c) [(G*M*m)/R]1/2 d) [(2*G*M)/R]1/2
An object of mass m is launched from a planet of mass M and radius R. a)Derive and enter an expression for the minimum launch speed needed for the object to escape gravity, i.e. to be able to just reach r = ∞.  b)Calculate this minimum launch speed (called the escape speed), in meters per second, for a planet of mass M = 6 × 1023 kg and R = 76 × 104 km.
D Gm₁m₂ Fg KE = mv², Ug = - 2πr , ac = =²₁, v = ²7₁ T Gm₁m₂ GM g = G, Vesc = 2GM R , E = KE + Ug, G = 6.674 x 10-¹1 Nm²/kg² Problem 1: You are the science officer on a visit to a distant solar system. Prior to landing on a planet you measure its radius to be 9 x 106 m and its rotation period to be 22.3 hours. You have previously determined that the planet orbits 2.2 x 10¹¹ m from its star with a period of 402 days (3.473 x 107 sec). Once on the surface you find that the free-fall acceleration is 12.2 m/sec². a) What is the mass of the planet? Answer: 1.5 x 1025 kg. b) What is the mass of the star? Answer: 5.2 x 1030 kg.
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