Question

Suppose that the tension is doubled for a string on which a wave is propagated. How will the velocity of the 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: (c) It will be multiplied by √2.

Step-by-step solution

Write down the given information and the formula for wave velocity

We know that the initial tension is \(T\) and it is doubled to \(2T\). The wave velocity is given by the formula: \(v = \sqrt{\frac{T}{\mu}}\).

Calculate the initial wave velocity

Using the formula, the initial wave velocity \(v_1\) can be calculated as: \(v_1 = \sqrt{\frac{T}{\mu}}\).

Calculate the new wave velocity after tension is doubled

When the tension is doubled, it becomes \(2T\). The new wave velocity \(v_2\) can be calculated as: \(v_2 = \sqrt{\frac{2T}{\mu}}\).

Find the ratio of the new wave velocity to the initial wave velocity

Divide \(v_2\) by \(v_1\) to find the ratio: \(\frac{v_2}{v_1} = \frac{\sqrt{\frac{2T}{\mu}}}{\sqrt{\frac{T}{\mu}}}\).

Simplify the expression

We simplify the expression: \(\frac{v_2}{v_1} = \sqrt{\frac{2T}{T}} = \sqrt{2}\).

Match the result with the given options

The velocity of the wave is multiplied by \(\sqrt{2}\). This matches with option (c). The correct answer is: (c) It will be multiplied by \(\sqrt{2}\).

Key concepts

Wave Propagation

Understanding how waves move through different media is an essential concept in physics. Wave propagation refers to the motion of waves as they travel through a medium. It’s not just about the movement itself, but also about how the properties of the medium affect the speed and form of the wave.

For instance, when dealing with mechanical waves, such as those traveling through a string, the velocity of wave propagation is determined by the tension of the string and the mass per unit length, or linear density, of the string. The formula \( v = \sqrt{\frac{T}{\mu}} \) encapsulates this relationship, where \( v \) is the wave velocity, \( T \) is the tension in the string, and \( \mu \) is the linear density.

When a wave is propagated on a string, and the tension is altered, the velocity of the wave changes accordingly. If the tension is increased, the wave travels faster; if the tension is decreased, the wave travels slower. This is because a tighter string is less resistant to the energy that moves the wave, resulting in higher speed.

Tension in Strings

Tension in strings is a fundamental concept that can affect wave propagation in systems such as musical instruments or bridges. In the context of waves on a string, tension refers to the force that is applied along the length of the string to stretch it. This force influences how wave energy is transmitted through the string.

A string under more tension is like a tight rubber band - it responds more promptly to disturbances and snaps back more quickly, which means that any wave traveling along it will move at a higher velocity. The relationship between tension and wave velocity is represented by the formula \( v = \sqrt{\frac{T}{\mu}} \) discussed earlier.

If we consider an exercise where the tension of a string is doubled, as in our example, the wave velocity would be affected as described by the formula \( v_2 = \sqrt{\frac{2T}{\mu}} \) after the change in tension. This clearly shows that wave velocity is directly proportional to the square root of the tension.

Physics Problem Solving

Physics problem solving is a systematic approach that helps to unpack complex scientific questions and reach concrete conclusions. First and foremost, it involves understanding the problem at hand and identifying the relevant physics principles that apply. In our example, the knowledge of how tension affects wave velocity was critical.

The next step usually involves translating this understanding into a mathematical representation, which may be a formula or an equation. The formula for wave velocity, \( v = \sqrt{\frac{T}{\mu}} \) as used in the example, is such a representation.

Following this, problem solvers will perform calculations as necessary, and interpret these results in the context of the given problem. In the last steps of our example, we compared the initial and new wave velocities to understand the relationship between the two. Simplifying the ratio led us to the final step, where we matched our finding that wave velocity is multiplied by \(\sqrt{2}\) when the tension in the string is doubled, to one of the multiple-choice options.