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Midterm 2

The second midterm exam is next Tuesday, 9 March 1999, 11:45 to 1:15. The exam is closed book. Bring a calculator.

The exam will focus on material from chapters 4 and 7, and includes the review of vector algebra. Of course these chapters build on material from chapters 2 and 3, so that material is very relevant.

Review

Chapter 2: Newtonian Mechanics, Rectilinear Motion of a Particle

• F = dp/dt = ma = m dv/dt = m d²x/dt² 
• Kinetic energy: T = 1/2 mv²
• Work: W = òFdx = DT
• Potential energy and potential functions.
• Solving problems with various types of forces:
  • constant forces
  • spring forces

Chapter 3: Oscillations

• General solution of the SHO equation.
• Amplitude, phase, kinetic and potential energy for a SHO
• Small oscillations about a minimum for an arbitrary potential.

Chapter 4: Motion in 2 and 3 Dimensions

Before proceeding into chapter 4, we reviewed vectors and vector algebra, and introduced some rules for differentiating vectors. Then we applied these skills to Newton's laws expressed in vector notation and applicable to problems involving more than 1 dimension.

• F = dp/dt = ma = m dv/dt  and   F₁₂ = -F₂₁.
• T = 1/2 mv·v = 1/2 mv²
• W = òF·ds = òF·v dt = DT
• A force is conservative if Ñ×F = 0. This means that a potential energy function, ÑV, exists such that F = V.
• A conservative force conserves mechanical energy: E = V + T = constant.
• Energy graphs: a plot of potential energy versus position and showing a line representing the total energy of the system. The endpoints or limits of allowed motion are found at the points where the potential energy curve crosses the total energy line.
• Force field diagrams and equipotential contours.
• Problems with forces of the separable type, especially the 2 special cases:
  • Problems with motion under a constant force.
  • Two and three dimensional harmonic oscillators.

Chapter 7: Systems of Particles

Chapter 7 is devoted to derivations of the forms of Newton's second law that are applicable to systems of point-like particles. The principle results are

• The center of mass (CM) of a system is given by a mass weighted average of the positions of the particles.
• The motion of the CM of the system is determined by total external force F_ext = dp_cm/dt.
• If there are no external forces then the momentum of the CM is conserved.
• The total kinetic energy is composed of the kinetic energy of the CM plus the kinetic energy of the motion of individual particles relative to the CM.
• Now that we have an extended system, it makes sense to discuss angular momentum:
  • The rate of change of the total angular momentum equals the total torque.
  • The total angular momentum is composed of two parts: the angular momentum of the CM of the system about the origin of coordinates, and the angular momentum of the system about its CM (rotations).
  • L_cm = r_cm×p_cm 
  • L_rel = S_i r_i'×p_i'
• We also discussed collision problems, impulse, and variable mass motion.
  • In collisions, momentum is conserved.
  • In elastic collisions, mechanical energy is conserved. In inelastic collisions, mechanical energy is not conserved. The extent to which energy is not conserved is measured by the quantity of restitution, e.
  • Impulse is a way of averaging over a rapidly changing force.
  • When a system is gaining or losing mass, we apply a variation of Newton's second law:
    F_ext = m dv/dt - v dm/dt.

Good luck on the exam.

© 04 March 1999 R. Harr