Physics 315: Oscillations and Waves Homework 9: Solutions
opaque screen
lightray projection screen
slits
d/2
R optical axis
1. Let d be the spacing of the slits, the wavelength of the light illuminating them, and R the distance of the (cylindrical) proje
Physics 315: Oscillations and Waves Homework 4: Solutions
1. Consider the ith loop from the left in the circuit, which consists of the ith capacitor, the ith inductor, and the i + 1th capacitor. The current owing through the ith inductor is Ii . Thus, the
Physics 315: Oscillations and Waves Homework 8: Solutions
1. Let x measure vertical distance upward, such that the bottom of the rope is at x = 0, and the top at x = L. Let the rope be of uniform mass per unit length . The weight of that section of the ro
Physics 315: Oscillations and Waves Homework 7: Solutions
1. Consider F (k ) = = 1 2 1 2 x2 exp 2 2 x
exp(i k x) dx (1)
x2 exp + i k x dx. 2 2 x
Now, completing the square,
2 4 x2 (x i x k )2 x k 2 + ikx = , 2 2 2 x 2 x 2 (x i x k )2 exp dx, 2 2 x 2 w
Physics 315: Oscillations and Waves Homework 6: Due in class on Wednesday, Oct. 28th
1. Write the traveling wave (x, t) = A cos(k x t) as a superposition of two standing waves. Write the standing wave (x, t) = A cos(k x) cos( t) as a superposition of two
Physics 315: Oscillations and Waves Homework 6: Solutions
1. Making use of the identity cos( ) cos cos + sin sin , we can write the traveling wave solution (x, t) = A cos(k x t) in the form (x, t) = A cos(k x) cos( t) + A sin(k x) sin( t). But, this is ju
Physics 315: Oscillations and Waves Homework 5: Solutions
1. Consider a small element of the string lying between x and x + dx. The mass of the element is dx. Furthermore, the elements transverse velocity is y (x, t)/t. Hence, the kinetic energy of the el
Physics 315: Oscillations and Waves Homework 7: Due in class on Wednesday, Nov. 4th
1. Suppose that x2 F (x) = exp 2 2 x Demonstrate that 1 F (k ) 2
.
F (x) e
ikx
dx =
1 k2 exp 2 2 2 k 2 k
,
where i is the square-root of minus one, and k = 1/x . [Hint: Yo
Physics 315: Oscillations and Waves Homework 10: Due in class on Wednesday, Dec. 2nd
1. (a) Consider the geometric series S=
n=0,N 1
zn,
where z is a complex number. Demonstrate that 1 zN S= . 1z (b) Suppose that z = e i , where is real. Employing the wel
Physics 315: Oscillations and Waves Homework 8: Due in class on Friday, Nov. 13th
1. A uniform rope of mass per unit length and length L hangs vertically. Determine the tension T in the rope as a function of height from the bottom of the rope. Show that t
Physics 315: Oscillations and Waves Homework 5: Due in class on Wednesday, Oct. 7th
1. Consider a uniform string of length l, tension T , and mass per unit length which is stretched between two immovable walls. Show that the total energy of the string, wh
Physics 315: Oscillations and Waves Homework 10: Solutions
1. (a) Let S=
n=0,N 1
z n = 1 + z + + z N 2 + z N 1 .
(1)
It follows that z S = z + z 2 + + z N 1 + z N . Thus, S z S = 1 zN , or 1 zN S= . 1z (3) (4) (2)
(b) Let z = e i . It follows that e i N /
Physics 315: Oscillations and Waves Homework 1: Solutions
1. The body will y o the diaphragm whenever the diaphragms downward acceleration exceeds the acceleration, g , due to gravity. Hence, as the frequency increases, the body will rst y o the diaphragm
Physics 315: Oscillations and Waves Homework 9: Due in class on Wednesday, Nov. 25th
1. An interference experiment employs two narrow parallel slits of separation 0.25 mm, and monochromatic light of wavelength = 500 nm. Estimate the minimum distance that
Physics 315: Oscillations and Waves Homework 4: Due in class on Wednesday, Sept. 30th
1. The gure below shows the left and right extremities of a linear LC network consisting of N identical inductors of inductance L, and N + 1 identical capacitors of capa
Physics 315: Oscillations and Waves Homework 3: Due in class on Wednesday, Sept. 23rd
1. Show that for a damped driven harmonic oscillator which is driven close to its resonant frequency x(t) X0 2 0 (0 ) 0 cos( t) + X0 sin( t). 4 (0 )2 + 2 4 (0 )2 + 2
2 0