Problem 16 - University Physics with Modern Physics

Question

With what tension must a rope with length 2.50 \(\mathrm{m}\) and mass 0.120 \(\mathrm{kg}\) be stretched for transverse waves of frequency 40.0 \(\mathrm{Hz}\) to have a wavelength of 0.750 \(\mathrm{m} ?\)

Step-by-Step Solution

Answer

43.2 N

Step 1: Calculate the wave speed

To calculate the wave speed, use the relation between wave speed, frequency, and wavelength given by: \( v = f \cdot \lambda \). Here, \( f = 40.0 \ \text{Hz} \) and \( \lambda = 0.750 \ \text{m} \). So, \( v = 40.0 \times 0.750 = 30.0 \ \text{m/s} \).

Step 2: Calculate the linear mass density of the rope

The linear mass density \( \mu \) is the mass per unit length of the rope and is given by: \( \mu = \frac{m}{L} \). Here, \( m = 0.120 \ \text{kg} \) and \( L = 2.50 \ \text{m} \) , so \( \mu = \frac{0.120}{2.50} = 0.048 \ \text{kg/m} \).

Step 3: Use wave speed to find the tension

The wave speed \( v \) on a stretched string is given by the equation \( v = \sqrt{\frac{T}{\mu}} \), where \( T \) is the tension in the rope. Rearrange this to solve for \( T \): \( T = \mu v^2 \). Using \( \mu = 0.048 \ \text{kg/m} \) and \( v = 30.0 \ \text{m/s} \), we find \( T = 0.048 \times (30.0)^2 = 43.2 \ \text{N} \).

Key Concepts

Wave Speed

Wave speed is an important concept when dealing with waves on strings or ropes. It indicates how fast a wave travels along the medium. The wave speed \( v \) is determined by the product of its frequency \( f \) and wavelength \( \lambda \) given by the equation:

• \( v = f \times \lambda \)

For example, if you know the frequency of a wave is 40 Hz (hertz) and its wavelength is 0.750 m (meters), you can easily find the wave speed:

• Calculate \( v = 40 \times 0.750 = 30 \) m/s (meters per second).

This tells you that the wave moves at a speed of 30 m/s through the rope. Understanding wave speed is crucial for determining other characteristics of waves like tension and linear mass density.

Linear Mass Density

Linear mass density, denoted by the symbol \( \mu \), is the mass per unit length of a rope or string. It plays a key role in determining the wave speed on the string, along with the tension applied to it. You can calculate linear mass density using the formula:

• \( \mu = \frac{m}{L} \)

Where \( m \) is the mass of the rope, and \( L \) is its length. For instance, if a rope has a mass of 0.120 kg and a length of 2.50 m, the linear mass density will be:

• \( \mu = \frac{0.120}{2.50} = 0.048 \) kg/m.

This value is the mass distributed over each meter of the rope, which helps in calculating the tension needed for wave propagation.

Transverse Waves

Transverse waves are a type of wave where the oscillations or particle movements occur perpendicular to the direction of the wave's travel. This is commonly seen in waves on a stretched rope or string.

• These waves consist of peaks and troughs.
• The speed at which these waves move is influenced by both the tension in the rope and its linear mass density.

To support transverse waves, a rope must be under tension. You can determine the tension using the wave speed equation:

• \( v = \sqrt{\frac{T}{\mu}} \), where \( T \) is the tension, and \( \mu \) is the linear mass density.

Rearranging gives \( T = \mu v^2 \). For a wave speed of 30 m/s, if \( \mu = 0.048 \) kg/m, then tension \( T \) becomes:

• \( T = 0.048 \times 30^2 = 43.2 \) N (Newtons).

This calculation shows that to produce and sustain transverse waves in the rope, a specific amount of tension is essential.