Write down the potential energy U(0) of a simple pendulum

A massless spring has unstretched length /0 and force constant k. One end is now attached to the ceiling and a mass m is hung from the other. The equilibrium length of the spring is now /1. (a) Write down the condition that determines 11. Suppose now the spring is stretched a further distance x beyond its new equilibrium length. Show that the net force (spring plus gravity) on the mass is F = kx. That is, the net force obeys Hooke's law, when x is the distance from the equilibrium position a very useful result, which lets us treat a mass on a vertical spring just as if it were horizontal. (b) Prove the same result by showing that the net potential energy (spring plus gravity) has the form U (x) = const ikx2.

The potential energy of two atoms in a molecule can sometimes be approximated by the Morse function, U (r) = ARe(R-01.5 1)2 where r is the distance between the two atoms and A, R, and S are positive constants with S << R. Sketch this function for 0 < r < cc. Find the equilibrium separation rip, at which U (r) is minimum. Now write r = ro x so that x is the displacement from equilibrium, and show that, for small displacements, U has the approximate form U = const kx2. That is, Hooke's law applies. What is the force constant k?

Write down the potential energy U(0) of a simple pendulum (mass m, length 1) in terms of the angle 0 between the pendulum and the vertical. (Choose the zero of U at the bottom.) Show that, for small angles, U has the Hooke's law form U(0) = 4k02, in terms of the coordinate 0. What is k?

An unusual pendulum is made by fixing a string to a horizontal cylinder of radius R, wrapping the string several times around the cylinder, and then tying a mass m to the loose end. In equilibrium the mass hangs a distance lo vertically below the edge of the cylinder. Find the potential energy if the pendulum has swung to an angle 0 from the vertical. Show that for small angles, it can be written in the Hooke's law form U = 1142 . Comment on the value of k.

In Section 5.2 we discussed four equivalent ways to represent simple harmonic motion in one dimension: x(t) = C C2ei" (I) = B1 cos(cot) + B2 sin(cot) (II) = A cos(cot 8) (HI) = Re Cei" (IV) To make sure you understand all of these, show that they are equivalent by proving the following implications: I II = III IV = I. For each form, give an expression for the constants (C1, C2, etc.) in terms of the constants of the previous form.

A mass on the end of a spring is oscillating with angular frequency co. At t = 0, its position is xo > 0 and I give it a kick so that it moves back toward the origin and executes simple harmonic motion with amplitude 2xo. Find its position as a function of time in the form (III) of Problem 5.5.

a) Solve for the coefficients B1 and B2 of the form (II) of Problem 5.5 in terms of the initial position xo and velocity vo at t = 0. (b) If the oscillator's mass is m = 0.5 kg and the force constant is k = 50 N/m, what is the angular frequency co? If xo = 3.0 m and vo = 50 m/s, what are B1 and B2? Sketch x(t) for a couple of cycles. (c) What are the earliest times at which x = 0 and at which = 0?

a) If a mass m = 0.2 kg is tied to one end of a spring whose force constant k = 80 N/m and whose other end is held fixed, what are the angular frequency co, the frequency f , and the period r of its oscillations? (b) If the initial position and velocity are xo = 0 and vo = 40 m/s, what are the constants A and 6 in the expression x(t) = A cos(cot 6)?

The maximum displacement of a mass oscillating about its equilibrium position is 0.2 m, and its maximum speed is 1.2 m/s. What is the period r of its oscillations?

The force on a mass m at position x on the x axis is F = Fo sink ax, where Fo and a are constants. Find the potential energy U(x), and give an approximation for U(x) suitable for small oscillations. What is the angular frequency of such oscillations?

You are told that, at the known positions x1 and x2, an oscillating mass m has speeds v1 and v2. What are the amplitude and the angular frequency of the oscillations?

Consider a simple harmonic oscillator with period r. Let ( f ) denote the average value of any variable f (t), averaged over one complete cycle: 1 f (f) = f (t) dt . 0 (5.103) Prove that (T) = (U) = 4E where E is the total energy of the oscillator. [Hint: Start by proving the more general, and extremely useful, results that (sin2(cot 6)) = (cos2 (cot 8)) = 2. Explain why these two results are almost obvious, then prove them by using trig identities to rewrite sine 6 and cost 0 in terms of cos(261).]

The potential energy of a one-dimensional mass m at a distance r from the origin is U(r)=Uo(L A211) for 0 < r < o , with U0, R, and X all positive constants. Find the equilibrium position r0. Let x be the distance from equilibrium and show that, for small x, the PE has the form U = const kx2. What is the angular frequency of small oscillations?

Consider a particle in two dimensions, subject to a restoring force of the form (5.21). (The two constants kx and k may or may not be equal; if they are, the oscillator is isotropic.) Prove that its potential energy is l(kx,2 kyy2). (5.104)

The general solution for a two-dimensional isotropic oscillator is given by (5.19). Show that by changing the origin of time you can cast this in the simpler form (5.20) with 6 = (Sy 6x. [Hint: A change of origin of time is a change of variables from t to t' = t to. Make this change and choose the constant to appropriately, then rename t' to be t.]

Consider a two-dimensional isotropic oscillator moving according to Equation (5.20). Show that if the relative phase is 6 = 7r/2, the particle moves in an ellipse with semimajor and semiminor axes A and Ay.

Consider the two-dimensional anisotropic oscillator with motion given by Equation (5.23). (a) Prove that if the ratio of frequencies is rational (that is, cox /coy = p I q where p and q are integers) then the motion is periodic. What is the period? (b) Prove that if the same ratio is irrational, the motion never repeats itself.

The mass shown from above in Figure 5.27 is resting on a frictionless horizontal table. Each of the two identical springs has force constant k and unstretched length 10. At equilibrium the mass rests at the origin, and the distances a are not necessarily equal to lo. (That is, the springs may already be stretched or compressed.) Show that when the mass moves to a position (x, y), with x and y small, the potential energy has the form (5.104) (Problem 5.14) for an anisotropic oscillator. Show that if a <10 the equilibrium at the origin is unstable and explain why.

Consider the mass attached to four identical springs, as shown in Figure 5.7(b). Each spring has force constant k and unstretched length lo, and the length of each spring when the mass is at its equilibrium at the origin is a (not necessarily the same as 10). When the mass is displaced a small distance to the point (x, y), show that its potential energy has the form Ik'r2 appropriate to an isotropic harmonic oscillator. What is the constant k' in terms of k? Give an expression for the corresponding force.

Verify that the decay parameter ,8 1,82 coo for an overdamped oscillator (p > coo) de-creases with increasing p. Sketch its behavior for wo < p < 00.

Verify that the function (5.43), x(t) = tofir, is indeed a second solution of the equation of motion (5.28) for a critically damped oscillator (,8 = (00).

a) Consider a cart on a spring which is critically damped. At time t = 0, it is sitting at its equilibrium position and is kicked in the positive direction with velocity vo. Find its position x(t) for all subsequent times and sketch your answer. (b) Do the same for the case that it is released from rest at position x = xo. In this latter case, how far is the cart from equilibrium after a time equal to ro = 2n/wo, the period in the absence of any damping?

A damped oscillator satisfies the equation (5.24), where Fdmp = b1 is the damping force. Find the rate of change of the energy E = 4mi2 + 4kx2 (by straightforward differentiation), and, with the help of (5.24), show that dE/dt is (minus) the rate at which energy is dissipated by Fdrnp.

In our discussion of critical damping (p = 600), the second solution (5.43) was rather pulled out of a hat. One can arrive at it in a reasonably systematic way by looking at the solutions for /3 < too and carefully letting /3 coo, as follows: For /3 < wo, we can write the two solutions as x1(t) = e-Pt cos(wit) and x2(t) = e-13` sin(wit).' Show that as p wo, the first of these approaches the first solution for critical damping, x1(t) = e-Pt . Unfortunately, as /3 wo, the second of them goes to zero. (Check this.) However, as long as /3 wo, you can divide x2(t) by col and you will still have a perfectly good second solution. Show that as p wo, this new second solution approaches the advertised to-13`.

Consider a damped oscillator with p < No. There is a little difficulty defining the "period" r1 since the motion (5.38) is not periodic. However, a definition that makes sense is that r1 is the time between successive maxima of x (t). (a) Make a sketch of x (t) against t and indicate this definition of r on your graph. Show that ri = 27r/co1. (b) Show that an equivalent definition is that r1 is twice the time between successive zeros of x(t). Show this one on your sketch. (c) If /3 = w012, by what factor does the amplitude shrink in one period?

An undamped oscillator has period ro = 1.000 s, but I now add a little damping so that its period changes to ri = 1.001 s. What is the damping factor 8? By what factor will the amplitude of oscillation decrease after 10 cycles? Which effect of damping would be more noticeable, the change of period or the decrease of the amplitude?

As the damping on an oscillator is increased there comes a point when the name "oscillator" seems barely appropriate. (a) To illustrate this, prove that a critically damped oscillator can never pass through the origin x = 0 more than once. (b) Prove the same for an overdamped oscillator.

A massless spring is hanging vertically and unloaded, from the ceiling. A mass is attached to the bottom end and released. How close to its final resting position is the mass after 1 second, given that it finally comes to rest 0.5 meters below the point of release and that the motion is critically damped?

An undamped oscillator has period ro = 1 second. When weak damping is added, it is found that the amplitude of oscillation drops by 50% in one period ri. (The period of the damped oscillations is defined as the time between successive maxima, ri = 27r/w1. See Problem 5.25.) How big is /3 compared to coo? What is r1?

The position x(t) of an overdamped oscillator is given by (5.40). (a) Find the constants C1 and C2 in terms of the initial position xo and velocity vo. (b) Sketch the behavior of x (t) for the two cases that vo = 0 and that x0 = 0. (c) To illustrate again how mathematics is sometimes cleverer than we (and check your answer), show that if you let /3 0, your solution for x (t) in part (a) approaches the correct solution for undamped motion.

Computer] Consider a cart on a spring with natural frequency wo = 2n, which is released from rest at xo = 1 and t = 0. Using appropriate graphing software, plot the position x (t) for 0 < t < 2 and for damping constants ,8 = 0, 1, 2, 4, 6, 27r, 10, and 20. [Remember that x(t) is given by different formulas for ,8 < we, = coo, and /3 > cool

Computer] Consider an underdamped oscillator (such as a mass on the end of a spring) that is released from rest at position xo at time t = 0. (a) Find the position x(t) at later times in the form x(t) = e-fit[B cos(wit) + B2 sin(w1t)]. That is, find B1 and B2 in terms of xo. (b) Now show that if you let 13 approach the critical value wo, your solution automatically yields the critical solution. (c) Using appropriate graphing software, plot the solution for 0 < t < 20, with xo = 1, wo = 1, and /5 = 0, 0.02, 0.1, 0.3, and 1.

The solution for x (t) for a driven, underdamped oscillator is most conveniently found in the form (5.69). Solve that equation and the corresponding expression for i, to give the coefficients B1 and B2 in terms of A, 8, and the initial position and velocity xo and vo. Verify the expressions given in (5.70).

Suppose that you have found a particular solution xp(t) of the inhomogeneous equation (5.48) for a driven damped oscillator, so that Dxp = f in the operator notation of (5.49). Suppose also that x (t) is any other solution, so that Dx = f . Prove that the difference x xp must satisfy the corresponding homogeneous equation, D(x xp) = 0. This is an alternative proof that any solution x of the inhomogeneous equation can be written as the sum of your particular solution plus a homogeneous solution; that is, x = xp xh.

This problem is to refresh your memory about some properties of complex numbers needed at several points in this chapter, but especially in deriving the resonance formula (5.64). (a) Prove that any complex number z = x iy (with x and y real) can be written as z = re where r and 0 are the polar coordinates of z in the complex plane. (Remember Euler's formula.) (b) Prove that the absolute value of z, defined as Izi = r, is also given by 1z12 = zz*, where z* denotes the complex conjugate of z, defined as z* = x iy. (c) Prove that z* = (d) Prove that (zw)* = z*w* and that (1/z)* = 1/z*. (e) Deduce that if z = a/ (b ic), with a, b, and c real, then 1z12 = a2/(b2 + C2).

Computer] Repeat the calculations of Example 5.3 (page 185) with all the same parameters, but with the initial conditions xo = 2 and vo = 0. Plot x(t) for 0 < t < 4 and compare with the plot of Example 5.3. Explain the similarities and differences.

[Computer] Repeat the calculations of Example 5.3 (page 185) but with the following param-eters w = 27, coo = 0.25w, /3 = 0.2wo, fo = 1000 and with the initial conditions xo = 0 and vo = 0. Plot x(t) for 0 < t < 12 and compare with the plot of Example 5.3. Explain the similarities and differences. (It will help your explanation if you plot the homogeneous solution as well as the complete solution homogeneous plus particular.)

[Computer] Repeat the calculations of Example 5.3 (page 185) but take the parameters of the system to be w = coo = 1, /3 = 0.1, and fo = 0.4, with the initial conditions xo = 0 and vo = 6 (all in some apppropriate units). Find A and 3, and then B1 and B2, and make a plot of x (t) for the first ten or so periods.

[Computer] To get some practice at solving differential equations numerically, repeat the calculations of Example 5.3 (page 185), but instead of finding all the various coefficients just use appropriate software (for example, the NDSolve command of Mathematica) to solve the differential equation (5.48) with the boundary conditions xo = vo = 0. Make sure your graph agrees with Figure 5.15.

Consider a damped oscillator, with fixed natural frequency coo and fixed damping constant 8 (not too large), that is driven by a sinusoidal force with variable frequency co. Show that the amplitude of the response, as given by (5.71) is maximum when co = 2/32. (Note that so long as the resonance is narrow this implies w ti wo.)

We know that if the driving frequency w is varied, the maximum response (A2) of a driven damped oscillator occurs at w ti wo (if the natural frequency is wo and the damping constant p c 0 o) . Show that A2 is equal to half its maximum value when w = coo 8, so that the full width at half maximum is just 2,8. [Hint: Be careful with your approximations. For instance, it's fine to say w + coo ti 2wo, but you certainly mustn't say w coo = 0.]

A large Foucault pendulum such as hangs in many science museums can swing for many hours before it damps out. Taking the decay time to be about 8 hours and the length to be 30 meters, find the quality factor Q.

When a car drives along a "washboard" road, the regular bumps cause the wheels to oscillate on the springs. (What actually oscillates is each axle assembly, comprising the axle and its two wheels.) Find the speed of my car at which this oscillation resonates, given the following information: (a) When four 80-kg men climb into my car, the body sinks by a couple of centimeters. Use this to estimate the spring constant k of each of the four springs. (b) If an axle assembly (axle plus two wheels) has total mass 50 kg, what is the natural frequency of the assembly oscillating on its two springs? (c) If the bumps on a road are 80 cm apart, at about what speed would these oscillations go into resonance?

Another interpretation of the Q of a resonance comes from the following: Consider the motion of a driven damped oscillator after any transients have died out, and suppose that it is being driven close to resonance, so you can set w = co.. (a) Show that the oscillator's total energy (kinetic plus potential) is E = Zmw2A2. (b) Show that the energy A Edis dissipated during one cycle by the damping force Fdmp is 2n- ini3wA2. (Remember that the rate at which a force does work is Fv.) (c) Hence show that Q is 27r times the ratio E/ A Edis.

Consider a damped oscillator, with natural frequency and damping constant both fixed, that is driven by a force . (a) Find the rate P (t) at which F(t) does work and show that the average rate over any number of complete cycles is . (b) Verify that this is the same as the average rate at which energy is lost to the resistive force. (c) Show that as is varied is maximum when ; that is, the resonance of the power occurs at (exactly).

The constant term ac, in a Fourier series is a bit of a nuisance, always requiring slightly special treatment. At least it has a rather simple interpretation: Show that if f (t) has the standard Fourier series (5.82), then cio is equal to the average ( f ) of f (t) taken over one complete cycle.

In order to prove the crucial formulas (5.83)0[5.85) for the Fourier coefficients an and bn, you must first prove the following: c/2 r/2 if m = n 0 cos(ncot) cos(mcot) dJt = r/2 0 if m n. (5.105) (This integral is obviously r if m = n = 0.) There is an identical result with all cosines replaced by sines, and finally r/2 Jr/2 cos(ncot) sin (mot) dt = 0 for all integers n and m, (5.106) where as usual w = 27r/r. Prove these. [Hint: Use trig identities to replace cos(0) cos(0) by terms like cos(0 + 0) and so on.]

Use the results (5.105) and (5.106) to prove the formulas (5.83)(5.85) for the Fourier coeffi-cients an and bn. [Hint: Multiply both sides of the Fourier expansion (5.82) by cos(mwt) or sin(mwt) and then integrate from z/2 to r/2.]

[Computer] Find the Fourier coefficients an and bn for the function shown in Figure 5.28(a). Make a plot similar to Figure 5.23, comparing the function itself with the first couple of terms in the Fourier series, and another for the first six or so terms. Take fmax = 1. Figure 5.28 (a) Problem 5.49. (b) Problem 5.50

Computer] Find the Fourier coefficients an and bn for the function shown in Figure 5.28(b). Make a plot similar to Figure 5.23, comparing the function itself with the sum of the first couple of terms in the Fourier series, and another for the first 10 or so terms. Take fmax = 1.

You can make the Fourier series solution for a periodically driven oscillator a bit tidier if you don't mind using complex numbers. Obviously the periodic force of Equation (5.90) can be written as f = Re(g), where the complex function g is cx) g(t) = fneinwt. n=0 Show that the real solution for the oscillator's motion can likewise be written as x = Re(z), where z(t),Ecneinwt n=0 and cn = fn (02 n2 co2 + 2i finco 0 This solution avoids our having to worry about the real amplitude An and phase shift 5, separately. (Of course An and 8n are hidden inside the complex number Ca.)

[Computer] Repeat all the calculations and plots of Example 5.5 (page 199) with all the same parameters except that . Compare your results with those of the example.

[Computer] An oscillator is driven by the periodic force of Problem 5.49 [Figure 5.28(a)], which has period r = 2. (a) Find the long-term motion x (t), assuming the following parameters: natural period v, = 2 (that is, co. = 7), damping parameter p = 0.1, and maximum drive strength fn.. = 1. Find the coefficients in the Fourier series for x(t) and plot the sum of the first four terms in the series for 0 < t < 6. (b) Repeat, except with natural period equal to 3.

Let f (t) be a periodic function with period r. Explain clearly why the average of f over one period is not necessarily the same as the average over some other time interval. Explain why, on the other hand, the average over a long time T approaches the average over one period, as T --->- co.

To prove the Parseval relation (5.100), one must first prove the result (5.99) for the integral of a product of cosines. Prove this result, and then use it to prove the Parseval relation.

The Parseval relation as stated in (5.100) applies to a function whose Fourier series happens to contain only cosines. Write down the relation and prove it for a function 00 x(t) [A cos(ncot 5,,,) Bn sin(nwt 3n) ]. n=0

[Computer] Repeat the calculations that led to Figure 5.26, using all the same parameters except taking . Plot your results and compare your plot with Figure 5.26.