William Moebs; Samuel J. Ling; Jeff Sanny

Challenge Problems

144.

A copper wire has a radius of $200 μm$ 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: $(ρ = 8.96 \frac{g}{{cm}^{3}}, Y = 1.1 \times 10^{11} \frac{N}{m}) .)$

145.

A pulse moving along the x axis can be modeled as the wave function $y (x, t) = 4.00 m e^{− {(\frac{x + (2.00 m/s) t}{1.00 m})}^{2}} .$ (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 $t = 0.00 s$ and $t = 3.00 s$ to verify your answer in part (b).

146.

A string with a linear mass density of $μ = 0.0085 kg/m$ 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?

147.

Consider two wave functions $y_{1} (x, t) = A sin (k x - ω t)$ and $y_{2} (x, t) = A sin (k x + ω t + ϕ)$ . What is the wave function resulting from the interference of the two wave? (Hint: $sin (α \pm β) = sin α cos β \pm cos α sin β$ and $ϕ = \frac{ϕ}{2} + \frac{ϕ}{2}$ .)

148.

The wave function that models a standing wave is given as $y_{R} (x, t) = 6.00 cm sin (3.00 m^{−1} x + 1.20 rad)$ $cos (6.00 s^{−1} t + 1.20 rad)$ . 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 $t = 1.00 s$ to verify your answer.

149.

Consider two wave functions $y_{1} (x, t) = A sin (k x - ω t)$ and $y_{2} (x, t) = A sin (k x + ω t + ϕ)$ . The resultant wave form when you add the two functions is $y_{R} = 2 A sin (k x + \frac{ϕ}{2}) cos (ω t + \frac{ϕ}{2}) .$ Consider the case where $A = 0.03 m^{−1},$ $k = 1.26 m^{−1},$ $ω = π s^{−1}$ , and $ϕ = \frac{π}{10}$ . (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 $t = 1.00 s$ to verify your answer.