4. 17:34ul 4GA cool.ntu.edu.tw1 of 2Unit 9 Problem Set: Wave1. Anonuniform wire of length L and mass M has a variable linear mass density given by p = kx, where xis the distance from one end of the wire and k is a constant. (a) Show that M = kL?/2. (b) Showthat the time t required for a pulse generated at one end of the wire to travel to the other end is given byt = /8ML/9F, where F is the tension in the wire.2. A uniform circular hoop of string is rotating clockwise in the absence ofgravity. The tangential speed isFind the speed of waves on the string.(Note that the answer is independent of the radius of the hoop and the linearmass density of the string!)3. Two strings of linear mass density H1 and uz are knotted together at x = 0 and stretched to a tensionF.A wave y = Asink, (x - vịt) in the string of density Hi reaches the junction between the two strings,at which it is partly transmitted into the string of density Hz and partly reflected. Call these wavesBsink2(x – vzt) and Csink, (x + vịt), respectively. (a) Assuming that kzv2 = k1V1 = w and thatthe displacement of the knot arising from the incident and reflected waves is the same as that arising fromthe transmitted wave, show that A = B + C. (b) If it is assumed that both strings near the knot have thesame slope (why?) - that is, dy/dx in string 1 = dy/dx in string 2 - Show that C = A =kz+k1A- Under what conditions is C negative?4. The period of a pulsating variable star may be estimated by considering the star to be executing radiallongitudinal pulsations in the fundamental standing wave mode; that is, the radius varies periodically withthe time, with a displacement antinode at the surface. (a) Would you expect the center of the star to be adisplacement node or antinode? (b) By analogy with the open organ pipe, show that the period ofpulsation T is given by T = , where R is the equilibrium radius of the star and v, is the averagesound speed. (c) Typical white dwarf stars are composed of material with a bulk modulus of 1.33 x1022 Pa and a density of 1.0 x 1010 kg/m3. They have radii equal to 0.009 solar radius. What is theapproximate pulsation period of a white dwarf?5. The figure showswaves contained in a single instrument. It is usedto measure the speed V of a target objecttransmitter and receiver of(idealized as a flat plate) that is moving directlytoward the unit, by analyzing the waves reflectedTargetfrom it. (a) Apply the Doppler equations twice,first with the target as observer, and then with the target as a source, and show that the frequency fr ofthe reflected waves at the receiver is related to their source frequency f by fr = f ), where v isthe speed of the waves. (b) In a great many practical situations, V «v. In this case, show that theequation above becomes6. A uniform cable hangs vertically under its own weight. Show that the speed of waves on the cable is givenby v = Jyg, where y is the distance from the bottom of the cable.7. Show that the time it takes a wave to propagate up the cable in Problem 6 is t = L/g, wherecable length.is the

Question

Accepted Answer

Answered: Image /qna-images/answer/4398ae9e-d369-4891-9b29-3fd3119d2467.jpg