More Problems

Problem 1: 

A particle of mass M moves in a circular orbit of radius r about a fixed point under the influence of an attractive force F = -k/r³e_r.  If the potential energy of the particle is set to zero at infinity, what is the total energy of the particle?

Solution:

• Concept:
  Motion in a central potential, circular orbits
• Reasoning:
  F = -dU(r)/dr, E = T + U.
• Details of the calculation:
  F(r) = -|k|/r³, U(r) = -½k/r².
  Centripetal acceleration: mv²/r = |k|/r³,  T = ½mv² + ½k/r²,  T + U = 0.

Problem 2: 

A satellite is launched into a circular orbit of radius R around Earth.  A second satellite is launched into an orbit of radius 1.01 R.  Find the ration of the periods of the two satellites.

Solution:

• Concepts:
  Kepler's third law
• Reasoning:
  T² ∝ r³,  T = period, R = radius of orbit.
• Details of the calculation:
  (T₂/T₁)² = (R₂/R₁)³ = (1.01)³ = (1 + 0.01)³ ~ 1 + 0.03.
  T₂/T₁ = (1 + x)^½ ~ 1 + ½x = 1.015.

Problem 3: 

A satellite is in an elliptical orbit about Earth, with a maximum distance from the center of the earth of 4R_E, and a minimum distance of 3R_E, where R_E is the radius of Earth.  If g is the magnitude of the acceleration due to gravity at Earth's surface, what is the maximum speed of the satellite in terms of g and R_E?

Solution:

• Concept:
  Motion in a central potential
• Reasoning:
  For motion in a central potential energy and angular momentum are conserved.  These two equations are enough to answer the questions.
• Details of the calculation:
  The satellite has its maximum speed v_max at the minimum distance and its minimum speed v_min at the maximum distance.
  Angular momentum conservation:  v_max*3R_E = v_min*4R_E.
  Energy conservation: ½mv_min² - GMm/(4R_E) = ½mvmax² - GMm/(3R_E).
  v_max = (8gR_E/21)^½.

Problem 4:

A object of mass m orbits a planet of mass M.  The total energy of the object is E. 
When its radial distance from the center of the planet is GMm/(8|E|), the radial component of the velocity of the object is zero.  What is the eccentricity e of the orbit?

Hint:  For Kepler orbits p/r ∝ 1 + e cos(φ - φ₀).

Solution:

• Concepts:
  Kepler orbits
• Reasoning:
  All orbits with the same semi-major axis have the same total energy per unit mass and the same period.
  For a circular orbit mv²/R = GMm/R²,  T = ½GMm/R, U = -GMm/R,  E = -½GMm/R. 
  R = ½GMm/|E|is the relationship between R and E for a circular orbit.
• Details of the calculation:
  The semi-major axis of the orbit of the object is a = ½GMm/|E|.
  When R = GMm/(8|E|) = a/4, the radial component of the velocity of the object is zero the objects distance from the center of the planet is R_min.
  Therefore R_max = 2a - R_min = 2a - a/4 = 7a/4.
  R_min/R_max = (1 - e)/(1 + e) = 1/7.
  e = -3/4.

Problem 5:

A particle of mass m moves under the action of a central force whose potential energy function is U(r) = kr³,  k > 0. 
(a)  For what energy and angular momentum will the orbit be a circle of radius a about the origin?  What is the period of this circular motion? 
(b)  If the particle is slightly disturbed from this circular motion, what will be the period of small radial oscillations about r = a?

Solution:

• Concepts:
  A particle moving in a central potential
  U = U(r),  F = f(r)(r/r),  f(r) = -∂U/∂r.
  In a central potential the energy E and the angular momentum M are conserved.
  E = ½m(dr/dt)² + M²/(2mr²) + U(r) = ½m(dr/dt)² + U_eff(r),
  with U_eff(r) = M²/(2mr²) + U(r).
  md²r/dt² = -∂U_eff(r)/∂r.
• Reasoning:
  A potential energy function U(r) is given.  The particle moves under the action of a central force.
• Details of the calculation:
  (a)  For a circular orbit with radius a we need r = a, dr/dt = 0, d²r/dt² = 0.
  For a circular orbit at with radius a we therefore need ∂U_eff(r)/∂r|_a = 0.  
  ∂U_eff(r)/∂r =  3kr² - M²/(mr³).
  ∂U_eff(r)/∂r|_a = 0 --> 3ka² = M²/(ma³),  a⁵  =  M²/(3mk),  M = (3kma⁵)^½.
  E = U_eff(a) = ka³ + M²/(2ma²) = ka³ + 3kma⁵/(2ma²) = 5ka³/2.
  The period for the circular motion is τ = 2π/ω, ω = dΦ/dt = M/ma² = (3ka/m)^½.
  τ = 2π/(3ka/m)^½.
   
• (b)  For a stable orbit we need a restoring force.  Let r = a + ρ, r >> ρ.
  We need md²ρ/dt² = -αρ.
  md²ρ/dt² = -∂U_eff(ρ)/∂ρ = -(∂U_eff/∂ρ)|_ρ=0 - (∂²U_eff/∂ρ²)|_ρ=0ρ (series expansion).
  We need ∂²U_eff/∂ρ² > 0 at ρ = 0, or ∂²U_eff/∂r² > 0 at r = a.
  ∂U_eff(r)/∂r =  3kr² - M²/(mr³).
  ∂²U_eff(r)/∂r² = 6kr + 3M²/(mr⁴).
  ∂²Ueff/∂ρ²|_r=a = 6ka + 3M²/(ma⁴) = 6ka² + 3kma⁵/(ma⁴) = 9ka² > 0.
  The orbit with radius a is stable, α = 9ka².  The period of small radial oscillation about a is
  τ = 2π/ω_r,  ω_r = (α/m)^½ = (9ka²/m)^½.  τ = 2π/(9ka²/m)^½