Problem 27 - Physics

Question

A transverse wave is traveling on a string. The displacement \(y\) of a particle from its equilibrium position is given by \(y=(0.021 \mathrm{~m}) \sin (25 t-2.0 x) .\) Note that the phase angle \(25 t-2.0 x\) is in radians, \(t\) is in seconds, and \(x\) is in meters. The linear density of the string is \(1.6 \times 10^{-2} \mathrm{~kg} / \mathrm{m} .\) What is the tension in the string?

Step-by-Step Solution

Answer

The tension in the string is 2.5 N.

Step 1: Identify wave properties

The wave equation is \( y = (0.021 \, \text{m}) \sin(25t - 2.0x) \). From this equation, we can identify the wave number \( k = 2.0 \) and the angular frequency \( \omega = 25 \).

Step 2: Calculate wave speed

Wave speed \( v \) can be found using the formula \( v = \frac{\omega}{k} \). Here, \( \omega = 25 \) and \( k = 2.0 \). Thus, the speed \( v = \frac{25}{2.0} = 12.5 \, \text{m/s} \).

Step 3: Use the wave speed to find tension

The wave speed \( v \) on a string is related to the tension \( T \) and linear density \( \mu \) by the formula \( v = \sqrt{\frac{T}{\mu}} \). Rearrange this to find \( T = v^2 \mu \).

Step 4: Substitute values to find tension

Given \( v = 12.5 \, \text{m/s} \) and \( \mu = 1.6 \times 10^{-2} \, \text{kg/m} \), substitute these values into \( T = v^2 \mu \) to get \( T = (12.5)^2 \times (1.6 \times 10^{-2}) \). Calculate \( T = 2.5 \, \text{N} \).

Key Concepts

Transverse Wave

A transverse wave is a type of wave where particles of the medium move perpendicular to the direction of the wave's travel. Imagine a rope lying flat on the ground. When you give one end a quick upward flick, the wave travels along the rope, but the movement of the rope segments is vertical. This vertical movement is why it's called a transverse wave.

Transverse waves are common in everyday life. Light waves and waves on a string are typical examples. The equation for a transverse wave, such as the one given in the problem, can describe the displacement of any point on the wave at any time.

• The displacement in our wave equation, \(y=(0.021 \, \text{m}) \sin(25t-2.0x)\), is a sine function, reflecting how the wave oscillates over both time and space.
  
• The parts of the equation: amplitude, wave number, and angular frequency define the wave's characteristics.

Understanding these components helps in determining properties like wave speed and tension in the medium. Understanding how the wave moves is crucial for applications like transmitting signals through cables.

Wave Speed

Wave speed is how fast the wave travels through a medium. To find it, we can use the relationship between the wave's angular frequency and wave number. Angular frequency (\(\omega\)) and wave number (\(k\)) characterize how quickly the wave oscillates in time and space.

The formula for wave speed (\(v\)) is \(v = \frac{\omega}{k}\). In our example, the values are \(\omega = 25 \, \text{radians/s}\) and \(k = 2.0 \, \text{radians/m}\). So, the wave speed is \(v = \frac{25}{2.0} = 12.5 \, \text{m/s}\).

• This speed indicates how fast the energy is being transferred along the string.
• Wave speed is crucial in determining the tension of the string, as faster waves imply different forces acting within the medium.

Knowing wave speed can also help in applications like tuning musical instruments and designing effective communication systems.

Linear Density

Linear density (\(\mu\)), is a measure of how much mass is distributed along a unit length of string or material. It is expressed as mass per unit length (\( \text{kg/m} \)). In the context of waves, linear density plays a key role in determining wave speed and tension.

For our problem, the linear density is given as \(1.6 \times 10^{-2} \, \text{kg/m}\). This tells us how much mass of the string is present over each meter, which influences how the wave travels.

• A higher linear density means more mass, which affects how the wave moves and the tension needed to maintain the wave.
• The wave speed \(v\) and tension \(T\) are related by the equation \(v = \sqrt{\frac{T}{\mu}}\).

Understanding linear density helps in applications such as designing cables for specific loads and ensuring efficient signal transmission.

Angular Frequency

Angular frequency (\(\omega\)) measures how quickly a wave oscillates in time, usually expressed in radians per second. It is a cornerstone concept for describing oscillations and waves. The angular frequency relates to how many cycles happen per unit of time.

In the given wave equation, \(\omega = 25\), indicating that the sine wave completes 25 radians worth of phase in one second. This frequency directly impacts the wave speed and how the medium will respond to applied tension.

• Angular frequency is essential when calculating wave properties alongside wave number.
• It emphasizes temporal characteristics of the wave, such as how often a particle returns to its starting position.

Understanding angular frequency is crucial in fields like acoustics and electronics, where time-based waves are common. It allows engineers and physicists to predict and control wave behavior effectively.