A fellow student with a mathematical bent tells you that the wave function of a traveling wave on a thin rope is $y (x, t) = (2.30 mm) \cos [(16.98 r a d / m) x + (742 r a d / s) t]$ . 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.

검증된 단계별 안내

Step 1: Identify the wave function given in the problem: y(x, t) = 2.30 mm cos[(16.98 rad/m)x + (742 rad/s)t]. This function describes a traveling wave on a rope.

Step 2: Calculate the wave speed (v) using the wave number (k) and angular frequency (ω). The wave number k is 16.98 rad/m, and the angular frequency ω is 742 rad/s. Use the formula v = ω/k to find the wave speed.

Step 3: Determine the linear mass density (μ) of the rope. The linear mass density is the mass per unit length of the rope. Use the formula μ = mass/length, where the mass is 0.00338 kg and the length is 1.35 m.

Step 4: Calculate the tension (T) in the rope using the wave speed and linear mass density. The formula for tension is T = μv², where μ is the linear mass density and v is the wave speed calculated in Step 2.

Step 5: Calculate the average power (P) transmitted by the wave. Use the formula P = (1/2)μvω²A², where A is the amplitude of the wave (2.30 mm converted to meters), μ is the linear mass density, v is the wave speed, and ω is the angular frequency.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Wave Function

The wave function y(x, t) = 2.30mm cos[(16.98 rad/m)x + (742 rad/s)t] describes the displacement of a wave at position x and time t. It includes parameters like amplitude, wave number, and angular frequency, which are crucial for understanding wave behavior and calculating properties like speed and energy.

Tension in the Rope

Tension in the rope is a force that affects wave propagation speed. It can be calculated using the formula v = sqrt(T/μ), where v is the wave speed, T is the tension, and μ is the linear mass density (mass per unit length). Understanding tension is essential for determining how fast waves travel along the rope.

Average Power Transmitted by the Wave

The average power transmitted by a wave is the rate at which energy is transferred along the medium. It can be calculated using P_avg = (1/2)μω^2A^2v, where μ is the linear mass density, ω is the angular frequency, A is the amplitude, and v is the wave speed. This concept helps in understanding the energy dynamics of wave motion.