1. A particle moves in a medium under the influence of a retarding force equal to -mk(v3+a2v), where k and a are positive constants. Show that for any value of the initial speed the particle will never move a distance greater than π/2ka and that the particle comes to rest only when the time approaches infinity.
2. Show directly that the time rate of change of the angular momentum about the origin for a projectile fired from the origin is equal to the moment of the gravitational force (its torque) about the origin.
3. A particle of mass m = 1 kg is subjected to a one-dimensional force
where k = 1 N/s and a = 0.5 s-1. If the particle is initially at rest, calculate and plot with the aid of a computer the position, speed, and acceleration of the particle as a function of time.
4. Consider a particle moving in the region x > 0 under the influence of the potential
where U0 = 1 J and a = 2 m. Plot the potential, find the equilibrium points, and determine whether they are maxima or minima.
5. Which of the following forces are conservative?
a. Fx = ayz + bx + c, Fy = axz + bz, Fz = axy + by
b. Fx = -ze-x, Fy = lnz, Fz = e-x + y/z
In these equations a, b, and c are constants. For the conservative forces, find the potential energy U.
Practice the material and concepts discussed in Chapter 2.