Chapter 4, Problem 4.3P

Let the initial position and speed of an overdamped, nondriven oscillator be x0 and v0, respectively. (a) Show that the values of the amplitudes A1 and A2 in Equation 3.44 have the values A1=2x0+v021 and A2=1x0+v021 where 1 = 2 and 2 = + 2. (b) Show that when A1 = 0, the phase paths of Figure 3-11 must be along the dashed curve given by x=2x, otherwise the asymptotic paths are along the other dashed curve given by x=1x. Hint: Note that 2 1 and find the asymptotic paths when t .

Consider the van der Waals potential U(r)=U0[( R 0 r)122( R 0 r)6] , used to model the potential energy function of two molecules, where the minimum potential is at r=R0 . Find the force as a function of r. Consider a small displacement R=R0+r and use the binomial theorem: (1+x)n=1+nx+n( n1)2!x2+n( n1)( n2)3!x3+ , to show that the force does approximate a Hooke’s law force.

Explain the difference between time-dependent and independent SchrÖdinger's equations.

Obtain the response of a linear oscillator to a step function and to an impulse function (in the limit τ → 0) for overdamping. Sketch the response functions.

Let's consider a simple pendulum, consisting of a point mass m, fixed to the end of a massless rod of length l, whose other end is fixed so that the mass can swing freely in a vertical plane. The pendulum's position can be specified by its angle Φ from the equilbrium position. Prove that the pendulum's potential energy is U(ϕ)=mgl(1−cosϕ). Write down the total energy E as a function of Φ and ϕ˙. Show that by differentiating E with respect to t you can get the equation of motion for Φ. Solve for Φ(t). If you solve properly, you should find periodic motion. What is the period of the motion?

Note: Because the argument of the trigonometric functions in this problem will be unitless, your calculator must be in radian mode if you use it to evaluate any trigonometric functions. You will likely need to switch your calculator back into degree mode after this problem.A massless spring is hanging vertically. With no load on the spring, it has a length of 0.24 m. When a mass of 0.59 kg is hung on it, the equilibrium length is 0.98 m. At t=0, the mass (which is at the equilibrium point) is given a velocity of 4.84 m/s downward. At t=0.32s, what is the acceleration of the mass? (Positive for upward acceleration, negative for downward)

Explain the term amplitude as it relates to the graph of asinusoidal function.

A body of mass m is suspended by a rod of length L that pivots without friction (as shown). The mass is slowly lifted along a circular arc to a height h. a. Assuming the only force acting on the mass is the gravitational force, show that the component of this force acting along the arc of motion is F = mg sin u. b. Noting that an element of length along the path of the pendulum is ds = L du, evaluate an integral in u to show that the work done in lifting the mass to a height h is mgh.

Show that the one-dimensional p(x) = Ae−ikx and three-dimensional expression p(x, y, z) = Ae−ikr is asolution to the Cartesian form of the Helmholtz equation, ∇2p + k2p = 0