Challenge Problems
A copper wire has a radius of 200μm and a length of 5.0 m. The wire is placed under a tension of 3000 N and the wire stretches by a small amount. The wire is plucked and a pulse travels down the wire. What is the propagation speed of the pulse? (Assume the temperature does not change: (ρ=8.96gcm3,Y=1.1×1011Nm).)
A pulse moving along the x axis can be modeled as the wave function y(x,t)=4.00me−(x+(2.00m/s)t1.00m)2. (a)What are the direction and propagation speed of the pulse? (b) How far has the wave moved in 3.00 s? (c) Plot the pulse using a spreadsheet at time t=0.00s and t=3.00s to verify your answer in part (b).
A string with a linear mass density of μ=0.0085kg/m is fixed at both ends. A 5.0-kg mass is hung from the string, as shown below. If a pulse is sent along section A, what is the wave speed in section A and the wave speed in section B?
Consider two wave functions y1(x,t)=Asin(kx−ωt) and y2(x,t)=Asin(kx+ωt+ϕ). What is the wave function resulting from the interference of the two wave? (Hint: sin(α±β)=sinαcosβ±cosαsinβ and ϕ=ϕ2+ϕ2.)
The wave function that models a standing wave is given as yR(x,t)=6.00cmsin(3.00m−1x+1.20rad) cos(6.00s−1t+1.20rad). What are two wave functions that interfere to form this wave function? Plot the two wave functions and the sum of the sum of the two wave functions at t=1.00s to verify your answer.
Consider two wave functions y1(x,t)=Asin(kx−ωt) and y2(x,t)=Asin(kx+ωt+ϕ). The resultant wave form when you add the two functions is yR=2Asin(kx+ϕ2)cos(ωt+ϕ2). Consider the case where A=0.03m−1, k=1.26m−1, ω=πs−1, and ϕ=π10. (a) Where are the first three nodes of the standing wave function starting at zero and moving in the positive x direction? (b) Using a spreadsheet, plot the two wave functions and the resulting function at time t=1.00s to verify your answer.