Question

Suppose that the tension is doubled for a string on which a standing wave is propagated. How will the velocity of the standing wave change? a) It will double. c) It will be multiplied by \(\sqrt{2}\). b) It will quadruple. d) It will be multiplied by \(\frac{1}{2}\).

Short answer

Answer: The velocity will be multiplied by √2.

Step-by-step solution

Understand the wave speed formula

Before calculating the change in velocity, it is essential to understand the wave speed formula. The formula for the speed of a wave on a string is given by: v = \(\sqrt{\frac{T}{\mu}}\) where v is the velocity of the wave, T is the tension on the string, and μ is the linear mass density of the string.

Analyze the change in tension

In this problem, we are given that the tension is doubled. This means that we have a new tension T' = 2T, where T' represents the doubled tension.

Calculate the new velocity

Now let's find the new velocity (v') for the wave with the doubled tension. Using the wave speed formula, we have: v' = \(\sqrt{\frac{T'}{\mu}}\) Since T' = 2T, we can substitute and get: v' = \(\sqrt{\frac{2T}{\mu}}\)

Compare the new and initial velocities

To find how the velocity of the standing wave changes, we need to compare the new velocity (v') with the initial velocity (v). We have: v' = \(\sqrt{\frac{2T}{\mu}}\) v = \(\sqrt{\frac{T}{\mu}}\) Divide v' by v to get: \(\frac{v'}{v}\) = \(\frac{\sqrt{\frac{2T}{\mu}}}{\sqrt{\frac{T}{\mu}}}\) = \(\sqrt{2}\)

Answer the question

We found that the velocity of the standing wave will be multiplied by \(\sqrt{2}\) when the tension is doubled. Therefore, the correct answer is (c) It will be multiplied by \(\sqrt{2}\).

Key concepts

Wave Speed Formula

Understanding how the velocity of a wave on a string is calculated is crucial for students dealing with wave mechanics. The wave speed formula is fundamental and is given by \[ v = \sqrt{\frac{T}{\mu}} \]where \( v \) is the velocity of the wave, \( T \) represents the tension in the string, and \( \mu \) symbolizes the linear mass density. In simpler terms, this equation tells us that the speed of a wave is proportional to the square root of the tension in the string and inversely proportional to the square root of the string's linear mass density.

This relationship is helpful in a variety of physics problems, especially those involving the study of waves on strings and in other mediums. When applied to a standing wave, it becomes a powerful tool to predict how changes in string tension, as in our exercise, would affect the wave's velocity.

Tension in Waves

Tension plays a pivotal role in the propagation of waves along a string or cable. It is, in essence, the force that stretches the material, providing the medium for the wave to travel on. As the tension increases, the medium becomes more taut, allowing waves to travel faster. Conversely, if the tension decreases, the medium becomes looser, and waves tend to slow down.

The direct relationship between tension and wave speed is visually observable. For example, when a musician tightens a guitar string (increasing tension), the pitch (which corresponds to wave frequency and speed) goes up. This observation aligns perfectly with the wave speed formula; as \( T \) increases, so does \( v \), assuming constant linear mass density. It's critical that students grasp the interplay between tension and wave velocity to accurately predict the behavior of waves under various tensions.

Standing Waves

Standing waves, often encountered in physics and engineering, are unique wave patterns that occur when two waves of the same frequency and amplitude interfere while traveling in opposite directions. They create nodes, points of no displacement, and antinodes, points of maximum displacement, that appear to be standing still, hence the name.

In the exercise, we discussed a standing wave on a string, which forms patterns of nodes and antinodes along the string's length. These patterns depend on the wave's frequency and the string's fixed length. The speed of the standing wave, influenced by both tension and linear mass density, affects the wavelengths and the standing wave's frequency. Recognizing standing waves and their characteristics helps in understanding resonance phenomena, musical instruments, and even quantum physics scenarios where similar principles apply.

Linear Mass Density

Linear mass density, denoted by \( \mu \), is the measure of mass per unit length of a material and is a key factor in the behavior of waves on strings. A higher linear mass density implies a heavier string and consequently a slower wave speed, assuming the tension remains constant. This can be understood when considering the inertia of the heavier string, which resists movement more than a lighter one.

In the context of our exercise, it's important to note that linear mass density doesn't change when the tension is altered, meaning its effect on wave speed is constant in such scenarios. Seeking a deep understanding of linear mass density enables students to predict how various strings or cables might carry waves differently, a concept valuable in fields like musical instrument design, civil engineering, and telecommunications.