A fellow student with a mathematical bent tells you that the wave function of a traveling wave on a thin rope is y(x, t)=2.30mm cos[(16.98 rad/m^)x+(742 rad/s)t]. Being more practical, you measure the rope to have a length of 1.35 m and a mass of 0.00338 kg. You are then asked to determine the following: (f) tension in the rope; (g) average power transmitted by the wave.

Verified step by step guidance

1. Step 1: Calculate the linear mass density ($\mu$) of the rope using the formula $\mu =\frac{m}{L}$, where $m$ is the mass of the rope and $L$ is its length.
2. Step 2: Identify the wave number ($k$) and angular frequency ($\omega$) from the wave function. The wave function given is $y(x,t)=2.30\text{ mm}\cos [(16.98\text{ rad/m})x+(742\text{ rad/s})t]$, where $k=16.98\text{ rad/m}$ and $\omega =742\text{ rad/s}$.
3. Step 3: Use the relationship between wave speed ($v$), wave number ($k$), and angular frequency ($\omega$) given by $v=\frac{\omega }{k}$ to find the wave speed.
4. Step 4: Calculate the tension ($T$) in the rope using the formula $T=\mu v^2$, where $v$ is the wave speed calculated in the previous step.
5. Step 5: Determine the average power transmitted by the wave using the formula $P=\frac{1}{2}\mu {\omega }^2{A}^{2}v$, where $A$ is the amplitude of the wave.

Key Concepts

Wave Function

The wave function describes the displacement of a wave at any point in space and time. In the given equation, y(x, t) = 2.30 mm cos[(16.98 rad/m)x + (742 rad/s)t], the amplitude is 2.30 mm, the wave number is 16.98 rad/m, and the angular frequency is 742 rad/s. Understanding the wave function is crucial for analyzing wave properties such as speed, frequency, and wavelength.

Tension in a Rope

Tension is the force exerted along the length of a rope or string when it is pulled tight. It plays a significant role in wave propagation, as the speed of a wave on a rope is influenced by the tension and the mass per unit length of the rope. The relationship can be expressed as v = √(T/μ), where T is tension and μ is mass per unit length, making it essential for calculating the tension in the rope.

Average Power of a Wave

The average power transmitted by a wave is the rate at which energy is transferred through a medium. For a traveling wave on a rope, the average power can be calculated using the formula P = (1/2)μω²A²v, where μ is mass per unit length, ω is angular frequency, A is amplitude, and v is wave speed. This concept is vital for understanding how much energy the wave carries as it travels along the rope.