Ch 16: Traveling Waves

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 16: Traveling Waves

Problem 59

Chapter 16, Problem 59

The string in FIGURE P16.59 has linear density μ. Find an expression in terms of M, μ, and θ for the speed of waves on the string.

Verified step by step guidance

Step 1: Begin by identifying the forces acting on the system. The block of mass M is on an inclined plane with angle θ, and the string is under tension due to the weight of the block. The tension in the string is caused by the component of the gravitational force acting along the incline.

Step 2: Write the expression for the tension in the string. The gravitational force acting on the block is given by F = M * g, where g is the acceleration due to gravity. The component of this force along the incline is T = M * g * sin(θ), which represents the tension in the string.

Step 3: Recall the formula for the speed of waves on a string. The speed of waves v is given by v = sqrt(T / μ), where T is the tension in the string and μ is the linear density of the string.

Step 4: Substitute the expression for tension into the wave speed formula. Replace T with M * g * sin(θ) in the formula for v, resulting in v = sqrt((M * g * sin(θ)) / μ).

Step 5: Simplify the expression to obtain the final formula for the speed of waves on the string. The speed of waves is v = sqrt((M * g * sin(θ)) / μ), which is expressed in terms of M, μ, and θ.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Wave Speed on a String

The speed of waves on a string is determined by the tension in the string and its linear density. The formula for wave speed (v) is given by v = √(T/μ), where T is the tension and μ is the linear density. Understanding how tension is affected by external forces, such as gravity acting on a mass on an incline, is crucial for deriving the wave speed in this scenario.

Linear Density (μ)

Linear density (μ) is defined as the mass per unit length of a string, typically expressed in kg/m. It plays a significant role in wave mechanics, as it influences how quickly waves can propagate through the medium. In this problem, knowing the linear density allows us to relate the mass of the string to its physical properties and the resulting wave speed.

Tension in the String

Tension in the string is the force exerted along the length of the string, which is crucial for wave propagation. In the context of the inclined plane, the tension can be derived from the gravitational force acting on the mass (M) and the angle (θ) of the incline. The vertical component of the gravitational force contributes to the tension, which directly affects the wave speed on the string.

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Energy & Power of Waves on Strings

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