Question

Calculate the speed of a transverse wave in a string of length 2.15 m and mass \(62.5 \mathrm{~g}\) under a tension of \(487 \mathrm{~N}\).

Short answer

The speed of the transverse wave in the string is approximately \(130.03 \, m/s\).

Step-by-step solution

Calculate the linear mass density

First, calculate the linear mass density (\(\mu\)) of the string. This can be done by dividing the mass of the string by its length. Since the mass is given in grams, it should be converted to kilograms by dividing by 1000. Thus, \(\mu = \frac{mass}{length} = \frac{62.5g}{2.15m} = \frac{62.5/1000 kg}{2.15m} = 0.02907 kg/m\)

Calculate the wave speed

Next, using the formula for wave speed (\(v = \sqrt{\frac{T}{\mu}}\)), find the speed of the transverse wave, substituting \(T = 487 N\), \(\mu = 0.02907 kg/m\) to get \(v = \sqrt{\frac{487N}{0.02907 kg/m}} = 130.03 m/s\).

Express the answer

Finally, express the answer. The speed of the transverse wave in the string is approximately 130.03 m/s.

Key concepts

Linear Mass Density

Understanding the linear mass density of a material is crucial when studying wave mechanics, particularly in strings and cables. It's a measure of mass per unit length and is represented by the symbol \( \mu \) (mu). To calculate \( \mu \) for a string, you simply divide the total mass of the string by its total length. It's essential to ensure the mass is in kilograms (kg) and length in meters (m) to keep the units consistent with the SI system.

During the exercise, the conversion from grams to kilograms is important because it aligns with the standard unit of mass in physics, preventing potential errors in calculations. \( \mu \) provides a base for further calculations regarding wave properties. Its value plays a pivotal role in determining the speed at which waves travel along the string under tension.

Remember, the linear mass density is not affected by the tension applied to the string; it only depends on the material and physical dimensions of the string itself.

Wave Speed Formula

The speed of a wave is a fundamental concept in physics, particularly in the study of mechanical waves such as transverse waves on a string. The wave speed formula represents how fast a wave travels through a medium, and for a string under tension, it takes a specific form: \( v = \sqrt{\frac{T}{\mu}} \), where \( v \) is the wave speed, \( T \) is the tension in the string, and \( \mu \) is the linear mass density of the string.

This formula showcases how the wave speed is directly proportional to the square root of the tension and inversely proportional to the square root of the linear mass density. High tension and low mass density will result in a faster wave speed, whereas low tension and high mass density will cause the wave to propagate more slowly. It is important to note that this relationship assumes the string is perfectly flexible and does not account for other factors like air resistance or internal damping.

Tension in Waves

Tension is a force that is applied along the length of an object, such as a string or a rope, and it has a crucial impact on wave behavior. In the context of waves on a string, tension determines the restoring force that wants to bring the string back to its equilibrium position after being disturbed.

Greater tension in the string results in a higher restoring force, which in turn makes waves travel faster along the string. It's essential to use the correct units when working with tension in wave calculations, typically Newtons (N) in the SI system. The strength of tension influences not only the speed of waves but also factors like wave frequency and amplitude, albeit the latter two require additional information beyond the scope of this exercise.

During the problem-solving process, always make sure to identify the given tension carefully and accurately incorporate it into the wave speed formula to arrive at an accurate calculation of wave velocity. By understanding tension's role, students can better predict and describe the dynamic behavior of waves in various physical systems.