(1 point) Consider the diffusion of a dye in a narrow tube 40 cm long whose ends are connected to...
(a) Consider an elastic string of length 10 whose ends are held fixed. The string is set in motion with no initial velocity from an initial postion J2/4 0<x<8 u(x,0) = { 0 8 << 10 Assuming the string is elastic enough to assume this initial configuration, i. Find the Fourier sine series for f (extended as an odd periodic function of period 20). ii. Assuming the propagation speed a = 2 solve the wave equation to find the displacement...
A plunger can be moved back and forth in a hollow tube that is L
= 150 cm long with two open ends. A stretched wire is placed near
the open end, as shown in the figure. The l = 50.0 cm long, 5.0
gram wire is fixed at both ends and oscillates in its fundamental
mode. By resonance (meaning that the frequency of the string equals
the frequency of the air column in the tube), the string sets the...
7. Find the solution of the heat conduction problem 100uzz = ut, 0 < x < 1, t > 0; u(0,t) 0, u1,t 0, t>0; In Problem 10, consider the conduction of heat in a rod 40 cm in length whose ends are maintained at 0°C for all t0. Find an expression for the temperature u(,t) if the initial temperature distribution in the rod is the given function. Suppose that a
Consider a 2 m long metal rod. The temperature u(z,t) at a point along the rod at any time t is found by solving the heat equation k where k is the material property. The left end of the rod ( 0) is maintained at 20°C and the right end is suddenly dipped into snow (0°C). The initial temperature distribution in the rod is given by u(x,0)- (i) Use the substitution u(z,t) ta,t)+20-10z to reduce the above problem to a...
3.1. Two metallic rods of length L = 40 cm and unknown linear mass density A are suspended from a support using lightweight strings of length s = 8 cm, as shown in the diagram (a) below. The rods are then connected to a circuit in such a way that same current I = 20 A passes through both rods, but flows in opposite directions. When connected in such a way, the rods move away from each other and at...
3.1. Two metallic rods of length L = 40 cm and unknown linear mass density 1 are suspended from a support using lightweight strings of length s = 8 cm, as shown in the diagram (a) below. The rods are then connected to a circuit in such a way that same current / = 20 A passes through both rods, but flows in opposite directions. When connected in such a way, the rods move away from each other and at...
A 200 Hz harmonic wave with an amplitude equal to 2 cm moves along a 40 m long string that has a mass of 120 grams and a tension of 50N. Considering that there is no energy lost, find: the power transmitted past a given point on the string and total average energy on a 20-m long segment of the string If the energy is lost in the process, the amplitude of the wave decreases as it travels along the...
1 point) Consider the initial value problem my" +cy' + ky-F(t), y' (0)-0 y(0)-0, modeling the motion of a spring-mass-dashpot system initially at rest and subjected to an applied force F(t), where the unit of force is the Newton (N) Assume that m = 2 kilograms, C-8 kilograms per second, k-80 Newtons per meter, and F(t) = 40 sin(6) Newtons. a. Solve the initial value problem. y0) help (formulas) b. Determine the long-term behavior of the system. ls lim y()...
(1 point) Consider the initial value problem my" cy' +ky = F(t), y(0) = 0, y'(0) = 0 modeling the motion of a spring-mass-dashpot system initially at rest and subjected to an applied force F(t), where the unit of force is the Newton (N). Assume that m 2 kilograms, 8 kilograms per second, k = 80 Newtons per meter, and the applied force in Newtons is c if 0tT/2, if t > /2 40 F(t)= a. Solve the initial value...
(1 point) Consider the initial value problem my + cy + ky = F(t), y0 = 0, y (0) = 0 modeling the motion of a spring-mass-dashpot system initially at rest and subjected to an applied force F(t), where the unit of force is the Newton (N). Assume that m = 2 kilograms, c = 8 kilograms per second, k = 80 Newtons per meter, and the applied force in Newtons is (20 if 0 <t< /2, F(t) = 30...