Problem 53 - University Physics with Modern Physics

Question

You must determine the length of a long, thin wire that is suspended from the ceiling in the atrium of a tall building. A 2.00-cm-long piece of the wire is left over from its installation. Using an analytical balance, you determine that the mass of the spare piece is 14.5 \(\mu\)g. You then hang a 0.400-kg mass from the lower end of the long, suspended wire. When a small-amplitude transverse wave pulse is sent up that wire, sensors at both ends measure that it takes the wave pulse 26.7 ms to travel the length of the wire. (a) Use these measurements to calculate the length of the wire. Assume that the weight of the wire has a negligible effect on the speed of the transverse waves. (b) Discuss the accuracy of the approximation made in part (a).

Step-by-Step Solution

Answer

The length of the wire is approximately 62.0 meters. The assumption about the negligible weight is reasonable given the data.

Step 1: Determine the Mass per Unit Length of the Spare Piece

First, calculate the mass per unit length of the spare piece of wire. Given that the mass of this 2.00-cm long wire is 14.5 \( \mu \)g, we convert these measurements into base units: 2.00 cm = 0.02 m and 14.5 \( \mu \)g = 14.5 \times 10^{-9} \text{ kg}. Then:\[\mu = \frac{14.5 \times 10^{-9} \text{ kg}}{0.02 \text{ m}} = 7.25 \times 10^{-7} \text{ kg/m}\]This is the mass per unit length of the wire.

Step 2: Determine the Tension in the Wire

The tension in the wire is due to the 0.400 kg mass hanging from it. The force exerted by this mass is its weight, which is calculated using:\[T = m \cdot g = 0.400 \text{ kg} \times 9.81 \text{ m/s}^2 = 3.924 \text{ N}\]where \( g = 9.81 \text{ m/s}^2\) is the acceleration due to gravity.

Step 3: Calculate the Speed of the Wave on the Wire

The speed of a transverse wave on a string is given by the formula:\[v = \sqrt{\frac{T}{\mu}}\]Substitute in the values for \(T\) and \(\mu\) to find:\\[v = \sqrt{\frac{3.924 \text{ N}}{7.25 \times 10^{-7} \text{ kg/m}}} = 2324.7 \text{ m/s}\]

Step 4: Calculate the Length of the Wire

Using the time it takes for the wave to travel the length of the wire (26.7 ms or 0.0267 s), and the wave speed calculated previously, the length \( L \) can be determined by rearranging the wave speed formula \( v = \frac{L}{t} \):\[L = v \times t = 2324.7 \text{ m/s} \times 0.0267 \text{ s} \approx 62.0 \text{ m}\]Thus, the length of the wire is approximately 62.0 meters.

Step 5: Evaluate Accuracy of Assumptions

The accuracy of the assumption that the weight of the wire is negligible in affecting the speed is dependent on how small the mass of the wire is compared to the tension created by the 0.400 kg mass. Our assumption is reasonable if the wire's mass is much less than the hanging mass. With our available data, the wire's weight is small relative to the tension, supporting the assumption.

Key Concepts

Wave Pulse on Wire

When you hear the term "wave pulse," imagine a single disturbance that travels through a medium. In this case, the medium is a thin wire. A wave pulse on a wire occurs when a brief disturbance creates a wave that moves along the wire. In our example scenario, a transverse wave pulse is sent up a suspended wire. Here are some essential points:

• The pulse's motion is perpendicular to the wire's length, hence the name "transverse."
• Transverse waves are crucial since they provide a way to measure unknown quantities such as the wire’s length.
• The speed and travel time of the wave pulse help calculate these measurements.

Understanding wave pulses enables us to apply physics concepts to real-world problems, such as determining the length of wires in buildings.

Mass per Unit Length

Mass per unit length is a simple but vital concept in understanding wave motion on a wire. It tells you how much mass exists in each unit of length of the wire. For our problem:

• The spare piece of wire, 2 cm long, has a mass of 14.5 µg, which tells us its mass per unit length.
• Converting these to standard units, we have a length of 0.02 m and a mass of 14.5 \( \times 10^{-9} \) kg.
• Dividing the mass by the length provides mass per unit length as \( 7.25 \times 10^{-7} \) kg/m.

This value is crucial for further calculations, like evaluating the wave's speed on the wire. Understanding mass per unit length helps in calculating how disturbances (like wave pulses) move through the material.

Transverse Wave Speed

The speed of a transverse wave on a wire depends on two main factors: the tension in the wire and its mass per unit length. The formula for transverse wave speed \( v \) is:\[v = \sqrt{\frac{T}{\mu}} \]where \( T \) is the tension and \( \mu \) is the mass per unit length. In our problem:

• The tension was found to be 3.924 N from the weight of a hanging mass.
• The mass per unit length (\( \mu \)) was previously calculated as \( 7.25 \times 10^{-7} \) kg/m.
• Substituting these into the formula, we find the wave speed to be 2324.7 m/s.

The wave speed is essential to determining how quickly information or energy travels along the wire.

Tension Calculation

Tension calculation is another fundamental step when dealing with physics problems on wires. Tension is essentially the pulling force exerted by a string or wire when an object is hung from it. To calculate the tension:

• Determine the weight of the object hanging from the wire.
• Apply the formula \( T = m \cdot g \), where \( m \) is the mass of the object (0.400 kg in this case) and \( g \) is the acceleration due to gravity (approximately 9.81 m/s²).
• The calculated tension in our example is 3.924 N.

Tension influences the speed of a wave pulse on the wire, making it a critical element of solving such physics problems.