Problem 17 - University Physics with Modern Physics

Question

The upper end of a 3.80-m-long steel wire is fastened to the ceiling, and a 54.0-kg object is suspended from the lower end of the wire. You observe that it takes a transverse pulse 0.0492 s to travel from the bottom to the top of the wire. What is the mass of the wire?

Step-by-Step Solution

Answer

The mass of the wire is approximately 0.34 kg.

Step 1: Calculate the Speed of the Pulse

The formula for speed is \( v = \frac{d}{t} \), where \( d \) is the distance and \( t \) is the time. Here, \( d = 3.80 \) m and \( t = 0.0492 \) s.\[ v = \frac{3.80}{0.0492} \approx 77.24 \text{ m/s} \]

Step 2: Relate Pulse Speed, Tension, and Mass Density

The speed \( v \) of a wave on a string is given by \( v = \sqrt{\frac{T}{\mu}} \), where \( T \) is the tension in the wire and \( \mu \) is the mass per unit length (linear density). We rearrange this to find \( \mu \):\[ \mu = \frac{T}{v^2} \]

Step 3: Calculate the Tension in the Wire

The tension \( T \) in the wire is equal to the weight of the object suspended, which can be calculated as \( T = mg \), where \( m \) is mass and \( g \) is acceleration due to gravity \( 9.81 \text{ m/s}^2 \).\[ T = 54.0 \times 9.81 = 529.74 \text{ N} \]

Step 4: Solve for Mass Density

Substituting the known values into the equation for \( \mu \):\[ \mu = \frac{529.74}{77.24^2} \approx 0.0889 \text{ kg/m} \]

Step 5: Calculate the Mass of the Wire

The mass of the wire \( M \) can be found using the linear density \( \mu \) and the total length of the wire. The formula is \( M = \mu \times L \), where \( L \) is the length of the wire (3.80 m).\[ M = 0.0889 \times 3.80 = 0.33782 \text{ kg} \]

Key Concepts

Transverse Waves

Transverse waves are waves where the displacement of the medium is perpendicular to the direction of wave propagation. This concept is integral to understanding wave mechanics because it explains how energy travels through different media.

• Examples of transverse waves include light waves or vibrations in a string.
• They are characterized by their crests (the highest points) and troughs (the lowest points).

Transverse waves in a wire, as in the exercise, allow us to study the motion and properties of waves such as speed, frequency, and amplitude. For the wire in question, a transverse pulse is created, and its travel time over a known distance helps us compute the wave's speed. This wave speed is essential in determining other factors like tension and linear density in the wire.
Understanding transverse waves enables you to comprehend how mechanical waves can convey energy without transporting matter. It also provides insights into the elastic properties of the medium in which they travel. This knowledge is crucial for analyzing various physical systems and engineering structures.

Linear Density

Linear density, denoted as \( \mu \), is a measure of mass per unit length. It's crucial in determining how a wave travels through a medium such as a wire.

• The formula \( \mu = \frac{m}{L} \) describes linear density, where \( m \) is mass and \( L \) is length.
• It's a key factor in calculating the speed of waves in a medium, especially when related to tension.

In the given problem, calculating linear density aids in understanding how the physical properties of the wire affect the behavior of the wave. The exercise demonstrates how linear density can be calculated using the tension and speed of the wave in the wire. By knowing the linear density, we can ascertain the mass of the wire based on its length and other dynamic properties. Comprehending linear density and its effects helps in designing and analyzing systems like musical instruments and suspension bridges, where wave mechanics play a vital role.

Tension in a Wire

The concept of tension in a wire refers to the force transmitted through a string, wire, or rope when it is pulled tight by forces acting from opposite ends.

• In physics, tension is typically measured in newtons (N).
• In our exercise, tension is determined by the weight of an object attached to the wire:
• \( T = mg \), where \( m \) is the mass and \( g \) is the acceleration due to gravity.

Tension is a critical element in determining the wave speed in a medium. For the given problem, it's essential to calculate the tension accurately, as it directly influences the wave's speed through the equation \( v = \sqrt{\frac{T}{\mu}} \). By knowing the tension and the wave speed, one can calculate the linear density, which is then used to ascertain the mass of the entire wire.
Tension is a widely applicable concept in various scientific and engineering disciplines, including material science and mechanical engineering. Understanding how tension affects wave motion can inform the design and analysis of structures or systems that require precision and reliability.