f(x) as in Figure 12.1. Hint: Integrate by parts and use (12.18) to evaluate Z

s = 3 cos 5t

s = 2 sin(4t 1)

s = 1 2 cos(t 8)

s = 5 sin(t )

s = 2 sin 3t cos 3t

s = 3 sin(2t + /8) + 3 sin(2t /8)

z = 5e it

z = 2e it/2

z = 2e it

z = 4ei(2t+3)

The charge q on a capacitor in a simple a-c circuit varies with time according to the equation q = 3 sin(120t + /4). Find the amplitude, period, and frequency of this oscillation. By definition, the current flowing in the circuit at time t is I = dq/dt. Show that I is also a sinusoidal function of t, and find its amplitude, period, and frequency.

Repeat Problem 11: (a) if q = Re 4e30it; (b) if q = Im 4e30it.

A simple pendulum consists of a point mass m suspended by a (weightless) cord or rod of length l, as shown, and swinging in a vertical plane under the action of gravity. Show that for small oscillations (small ), both and x are sinusoidal functions of time, that is, the motion is simple harmonic. Hint: Write the differential equation F = ma for the particle m. Use the approximation sin = for small , and show that = A sin t is a solution of your equation. What are A and ?

The displacements x of two simple pendulums (see Problem 13) are 4 sin(t/3) and 3 sin(t/4). They start together at x = 0. How long will it be before they are together again at x = 0? Hint: Sketch or computer plot the graphs.

As in Problem 14, the displacements x of two simple pendulums are x = 2 cos(t/2) and 3 sin(t/3). They are not together at t = 0; plot graphs to see when they are first together.

As in Problem 14, let the displacements be y1 = 3 sin(t/2) and y2 = sin t. The pendulums start together at t = 0. Make computer plots to estimate when they will be together again and then, by computer, solve the equation y1 = y2 for the root near your estimate.

Show that equation (2.10) for a wave can be written in all these forms: y = A sin 2 (x vt) = A sin 2 x t T = A sin x v t = A sin 2x 2ft = A sin 2 T x v t . Here is the wavelength, f is the frequency, v is the wave velocity, T is the period, and = 2f is called the angular frequency. Hint: Show that v = f.

y = 2 sin 2 3 (x 3t); t = 0, 1 4 , 1 2 , 3 4 ; x = 0, 1, 2, 3.

y = cos 2(x 1 4 t); t = 0, 1, 2, 3; x = 0, 1 4 , 1 2 , 3 4 .

y = 3 sin (x 1 2 t); t = 0, 1, 2, 3; x = 0, 1 2 , 1, 3 2 , 2.

Write the equation for a sinusoidal wave of wavelength 4, amplitude 20, and velocity 6. (See Problem 17.) Make computer plots of y as a function of t for x = 0, 1, 2, 3, and of y as a function of x for t = 0, 1 6 , 1 3 , 1 2 . If this wave represents the shape of a long rope which is being shaken back and forth at one end, find the velocity y/t of particles of the rope as a function of x and t. (Note that this velocity has nothing to do with the wave velocity v, which is the rate at which crests of the wave move forward.)

Do Problem 21 for a wave of amplitude 4, period 6, and wavelength 3. Make computer plots of y as a function of x when t = 0, 1, 2, 3, and of y as a function of t when x = 1 2 , 1, 3 2 , 2.

Write an equation for a sinusoidal sound wave of amplitude 1 and frequency 440 hertz (1 hertz means 1 cycle per second). (Take the velocity of sound to be 350 m/sec.)

The velocity of sound in sea water is about 1530 m/sec. Write an equation for a sinusoidal sound wave in the ocean, of amplitude 1 and frequency 1000 hertz.

Write an equation for a sinusoidal radio wave of amplitude 10 and frequency 600 kilohertz. Hint: The velocity of a radio wave is the velocity of light, c = 3108 m/sec.

f(x) = ( 3x2 + 2x3, 1

f(x) = ( 3x2 + 2x3, 1

f(x) = ( 1, 2 4

f(x) = 8 >< >: 1, 0 3

f(x) = ( 1 x/2, 0 2

f(x) as in Figure 12.1. Hint: Integrate by parts and use (12.18) to evaluate Z |g()| 2 d

f(x) and g() as in Problem 21.

f(x) and g() as in Problem 24a.

Show that if (12.2) is written with the factor 1/ 2 multiplying each integral, then the corresponding form of Parsevals theorem (12.24) is Z |f(x)| 2 dx = Z |g()| 2 d.

Starting with the symmetrized integrals as in Problem 34, make the substitutions p = 2p/h (where p is the new variable, h is a constant), f(x) = (x), g() = h/2 (p); show that then (x) = 1 h Z (p)e 2ipx/h dp, (p) = 1 h Z (x)e2ipx/h dx, Z |(x)| 2 dx = Z |(p)| 2 dp. This notation is often used in quantum mechanics.

Normalize f(x) in Problem 21; that is find the factor N so that R |Nf(x)| 2 = 1. Let (x) = Nf(x), and find (p) as given in Problem 35. Verify Parsevals theorem, that is, show that R |(p)| 2 dp = 1.