Conceptual Questions for Physics 311 (Theoretical Mechanics II)

• Chapter 8
  
  
• 1 :  What are Kepler's Three Laws ?
  
  
• 2 :  Explain why equal areas are swept out in equal time for any central force
  
  
• 3 :  What is reduced mass ? What is the reduced mass for the sun-earth system ? What is the reduced mass for positronium ? For the hydrogen atom ?
  
  
• 4 :  What is the equation of a conic section in polar coordinates ? When does the conic section describe a circle, an ellipse, a parabola, a hyperbola ?
  
  
• 5 :  Sketch the effective potential for a gravitational interaction of two particles. For what energies do closed orbits occur ? When are circular orbits possible ?
  
  
• 6 :  For what power law potential energy functions are closed non-circular orbits possible ?
  
  
• 7 :  What is a Hohmann transfer ? Does it change the angular momentum of the orbit ? How about the energy and eccentricity of the orbit ?
  
  
• 8 :  Explain how the "sling-shot" effect was used by the Voyager 2 spacecraft to travel to Uranus and Neptune.
  
  
• 9 :  What conditions are required for the effective potential for circular orbits to be stable ?
  
  
• 10:   For an elliptical orbit, what is the ratio of orbital speeds at r(max), the apogee and r(min), the perigee ?
  
  

• Chapter 9
  
  
• 1 :  What is the strong version of Newton's 3rd Law ? What is the center of mass for a system of point particles ? For a continuous object ?
  
  
• 2 :  How can the kinetic energy of a system be decomposed into two parts after introducing the center of mass (CM) ? What is the corresponding decomposition of the angular momentum ?
  
  
• 2 :  What is meant by an elastic collision ?
  
  
• 3 :  In the lab frame, a particle undergoes an elastic collision with another particle at rest. In the CM frame, what is the angle between outgoing particles ?
  
  
• 4 :  Sketch the effective potential for a gravitational interaction of two particles. For what energies do closed orbits occur ? For what energies are parabolic orbits possible ?
  
  
• 5 :  On the Voyager 2 missions to the outer planets of the solar system, were the spacecraft sent on a straight line path from Earth to Uranus and Neptune ?
  
  
• 6 :  Consider a collision between two particles, with one initially at rest. In the CM frame, what is the angle between the two outgoing particles ? (Explain).
  
  
• 7 :  In the lab frame, a particle ($m_1$) undergoes an elastic collision with another particle of much larger mass at rest. What is the relation between the LAB scattering angle and the CM scattering angle of $m_1$ ?
  
  
• 8 :  What is the dependence of Rutherford scattering differential cross angle on CM angle ? If the incoming particle is an electron, how does the cross section depend on the charge of the target ?
  
  

• Chapter 10
  
  
• 1 :  What is an inertial frame of reference ? (Explain or review p.53).
  
  
• 2 :  Is the earth an inertial frame of reference ?
  
  
• 3 :  What is the direction of rotation of cyclones in the northern hemisphere ? In the southern hemisphere ? Which way do toilets flush in the Northern and Southern hemispheres ?
  
  
• 4 :  How did Foucault demonstrate that the earth is rotating ?
  
  
• 5 :  Why did British gunners miss their targets by about 100 yards during WWI naval engagements near the Falkland Islands in the southern hemisphere ?
  
  

• Chapter 11
  
  
• 1 :  What is the moment of inertia tensor ? How is the angular momentum related to the moment of inertia tensor ? How is the kinetic energy related to the moment of inertia tensor ?
  
  
• 2 :  Two spheres have the same mass and diameter, but one is solid and the other is a hollow shell. Describe a non-destructive experiment to determine which is hollow and which is solid.
  
  
• 3 :  Draw a sketch of the bicycle wheel gyroscope. Explain with this diagram why it precesses.
  
  
• 4 :  Sketch the effective potential V(theta) for the symmetric top. For what energies, does nutation occur ? What is nutation ? What are the three kinds of nutation ?
  
  
• 5 :  For the bicycle wheel gyroscope, how does the angular velocity of precession depend on the angular velocity that the wheel is spinning about its own axis ? So when the bicycle slows down
  
  
• 6 :  What torque causes the "tippy-top" to flip over ?
  
  
• 7 :  Does the earth, our home planet, precess ? What are its typical periods of precession ?
  
  
• 8 :  What is the parallel axis theorem ? What is the moment of inertia of a sphere about an axis tangent to its surface ?
  
  
  • Chapter 12
    
    
  • 1 :  For a system of two identical pendula coupled by a spring, what are the two normal modes ? Which one has the higher frequency ? Why ?
    
    
  • 2 :  For a system of two identical springs coupled with a weak intermediate spring, describe the behaviour of the system ?
    
    
  • 3 :  For a symmetrical linear triatomic molecule, sketch the transverse and longitudinal normal modes
    
    
  • 4 :  For the loaded string with n=3 tied down at both ends, sketch the first three normal modes (see Figure 12-11)
    
    
  • 5 :  What are the normal modes of the system of 3 pendula coupled by a slightly yielding support bar ? (example 12.6)
    
    
  
  

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  • Chapter 13
    
    
  • 1 :  In what limit, does the loaded string become a continuous string ? Given the initial position q(x,0) and velocity qdot (x,0), explain how to find the normal modes of a continuous string.
    
    
  • 2:   The function sin(r \pi x /L) are said to be orthogonal in the interval 0,L. What does this mean ? How is this relevant to the normal modes of the string ?
    
    
  • 3 :  What is the wave equation for a continuous string in 1 dimension ?
    
    
  • 4 :  What is general solution for the wave equation in 1 dimension ?
    
    
  • 5 :  Consider a string tied down at one end. If a traveling wave propagates down the string towards the end that is tied down, describe what happens. Sketch the result.
    
    
  • 6 :  Consider two waves of equal amplitude but traveling in opposite directions. What happens when they meet ? Where do nodes occur ?
    
    
  • 7 :  How is the wave number related to the wave velocity and angular frequency ?
    
    
  • 8 :  What is phase velocity ? How does it differ from group velocity ? What is dispersion ?
    
    
  • 9 :  Consider a string with two densities rho_1 and rho_2. What boundary conditions must be satisfied by the wave at the interface ?
    
    
  • 10:   Is the wave number always real ? If not, what is the physical significance of imaginary k ?
    
    
  • 11:   What is a wave packet ?
    
    
  • 12:   Standing waves in an organ pipe have frequencies in the ratio 1:3:5. What can be said about the organ pipe ?
    
    
  
  

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  • Chapter 14
    
    
  • 1:   If two events are simultaneous in one reference frame, are they are simultaneous in all other references frames ?
    
    
  • 2:   Give a physical example of Lorentz contraction. Give an example of time dilation.
    
    
  • 3:   Give a physical example where E= m c^2 is relevant.
    
    
  
  

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  Tom Browder
  April 25, 2000