Question

A pulse is started at a time t = 0 along the +x direction on a long, taut string. The shape of the pulse at t = 0 is given by function f(x) with $f(x) = \begin{cases} \frac{x}{4} + 1 & \text{for } -4 < x \leq 0 \\ -x + 1 & \text{for } 0 < x < 1 \\ 0 & \text{otherwise} \end{cases}$ here f and x are in centimeters. The linear mass density of the string is 50 g/m and it is under a tension of 5N.

• A. The shape of the string is drawn at t = 0 then the area of the pulse enclosed by the string and the x-axis is 2.5 cm².
• B. The shape of the string is drawn at t = 0 then the area of the pulse enclosed by the string and the x-axis is 5 cm².
• C. The transverse velocity of the particle at x = 13 cm and t = 0.015 s will be - 250 cm/s
• D. The transverse velocity of the particle at x = 13 cm and t = 0.015 s will be 250 cm/s

Answer 1

Answer: (A), (C)

A, C

Answer 2

1. Calculate the area of the pulse at $t=0$ by integrating the given function $f(x)$ over its non-zero domain $(-4, 1)$. The integral is split into two parts corresponding to the piecewise definition of $f(x)$. The area is the sum of the areas of the two triangular regions formed by the pulse shape and the x-axis.

2. Calculate the speed of the pulse $v$ using the given tension $T$ and linear mass density $\mu$ with the formula $v = \sqrt{T/\mu}$. Ensure units are consistent. Convert the speed to cm/s as the position $x$ and function $f$ are given in cm.

3. The wave function at time $t$ is $y(x, t) = f(x - vt)$. The transverse velocity of a particle is $v_y = \frac{\partial y}{\partial t}$. Using the chain rule, $\frac{\partial y}{\partial t} = f'(x - vt) \frac{\partial}{\partial t}(x - vt) = -v f'(x - vt)$.

4. Evaluate the argument $u = x - vt$ at the given position $x = 13$ cm and time $t = 0.015$ s.

5. Determine the slope $f'(u)$ of the pulse shape at the calculated position $u$. This requires finding the derivative of the relevant part of the piecewise function $f(x)$.

6. Calculate the transverse velocity $v_y = -v f'(u)$ using the calculated speed and slope.

7. Compare the calculated area and transverse velocity with the values given in the options to identify the correct statements.