Problem 7 - College Physics

Question

\(\bullet\) One end of a horizontal rope is attached to a prong of an electrically driven tuning fork that vibrates at 120 \(\mathrm{Hz}\) . The other end passes over a pulley and supports a 1.50 \(\mathrm{kg}\) mass. The linear mass density of the rope is 0.0550 \(\mathrm{kg} / \mathrm{m}\) . (a) What is the speed of a transverse wave on the rope? (b) What is the wavelength? (c) How would your answers to parts (a) and (b) change if the mass were increased to 3.00 \(\mathrm{kg}\) ?

Step-by-Step Solution

Answer

Wave speed: 16.34 m/s; wavelength: 0.1362 m; increases: 23.06 m/s, 0.1922 m.

Step 1: Determine the tension in the rope

The tension in the rope is caused by the weight of the 1.50 kg mass hanging from it. To find the tension, use the formula for the force of gravity: \( T = m \cdot g \). The tension \( T \) is equal to \( 1.50 \, \mathrm{kg} \times 9.81 \, \mathrm{m/s^2} = 14.715 \, \mathrm{N} \).

Step 2: Calculate the speed of the wave (part a)

Use the formula for wave speed \( v \) on a string, which is \( v = \sqrt{\frac{T}{\mu}} \), where \( T \) is the tension and \( \mu = 0.0550 \, \mathrm{kg/m} \) is the linear mass density. Substitute the known values: \( v = \sqrt{\frac{14.715}{0.0550}} = 16.34 \, \mathrm{m/s} \).

Step 3: Determine the wavelength (part b)

The wavelength \( \lambda \) is related to wave speed \( v \) and frequency \( f \) by \( \lambda = \frac{v}{f} \). Given \( f = 120 \, \mathrm{Hz} \) and \( v = 16.34 \, \mathrm{m/s} \), substitute to find \( \lambda = \frac{16.34}{120} = 0.1362 \, \mathrm{m} \).

Step 4: Recalculate for increased mass (part c - tension)

With the mass increased to 3.00 kg, recalculate the tension: \( T = 3.00 \, \mathrm{kg} \times 9.81 \, \mathrm{m/s^2} = 29.43 \, \mathrm{N} \).

Step 5: Recalculate wave speed for increased mass

Use the updated tension to find the new wave speed: \( v = \sqrt{\frac{29.43}{0.0550}} = 23.06 \, \mathrm{m/s} \).

Step 6: Calculate new wavelength for increased mass

With the new speed \( v = 23.06 \, \mathrm{m/s} \), find the new wavelength: \( \lambda = \frac{23.06}{120} = 0.1922 \, \mathrm{m} \).

Key Concepts

Transverse Waves

Transverse waves are a type of wave where particles of the medium move perpendicular to the direction of the wave propagation. Picture a wave traveling along a string; the string itself moves up and down, while the wave travels left to right. This up-and-down movement is what defines it as transverse.

• Transverse waves are common in many applications, like waves on a string or water waves.
• Unlike longitudinal waves, such as sound waves, transverse waves involve particle motion that is perpendicular to the energy transport.
• In the given exercise, the wave travels along a rope, moved by the oscillations of a tuning fork at 120 Hz.

Understanding transverse waves is crucial for analyzing how energy transmits through various mediums. They allow us to determine properties like wave speed and wavelength, which are essential for practical calculations in physics.

Wave Speed

Wave speed is the speed at which the wave propagates through the medium. It is determined by the formula \( v = \sqrt{\frac{T}{\mu}} \), where \( T \) is the tension in the string and \( \mu \) is the linear mass density. This relationship is vital in calculating how fast disturbances travel through media like strings or ropes.

• For a rope with a linear density of 0.0550 kg/m and a tension due to a 1.50 kg mass, the speed is calculated to be 16.34 m/s.
• When the tension changes, as demonstrated with a mass increase to 3.00 kg, the wave speed increases to 23.06 m/s.

Wave speed is contingent on factors such as tension and medium density. In engineering and physics, knowing wave speed helps in designing instruments and understanding natural phenomena.

Tension in a String

Tension in a string is a force that is crucial in wave motion. It is caused by forces applied along the string, in this case, by weights hanging at the end. The tension determines how quickly waves can travel through the rope.

Use the formula \( T = m \cdot g \) to calculate tension, where:

• \( m \) is the mass (1.50 kg initially, then 3.00 kg),
• \( g \) is the gravitational acceleration (approximately 9.81 m/s²).

With the 1.50 kg mass, the tension is 14.715 N, while for the 3.00 kg mass, it increases to 29.43 N.

• Tension influences wave speed directly; higher tension usually leads to faster wave propagation.

Understanding tension not only helps in physics problems but is also practical in real-life applications like construction and musical instruments.

Wavelength Calculation

Wavelength is the distance between consecutive points of a wave that are in phase, such as two adjacent peaks. It's a fundamental property of waves associated with the wave's speed and frequency. The formula is \( \lambda = \frac{v}{f} \), linking wavelength \( \lambda \), speed \( v \), and frequency \( f \).

• For the initial conditions in the exercise, with a speed of 16.34 m/s and frequency of 120 Hz, the wavelength is 0.1362 m.
• After increasing the mass, leading to a speed of 23.06 m/s, the wavelength becomes 0.1922 m.

Wavelength is essential in analyzing wave behaviors and is used in various fields, from acoustics to optics. It helps in determining the nature and behavior of waves when their environments change.