Question

You wish to decrease the speed of a wave traveling on a string to half its current value by changing the tension in the string. By what factor must you decrease the tension in the string? a) 1 c) 2 e) none of the above b) \(\sqrt{2}\) d) 4

Short answer

Answer: (d) 4

Step-by-step solution

Write the given information

The current wave speed is \(v\) and we wish to decrease it to half its current value, which is \(\frac{1}{2}v\). The goal is to find the factor x by which the tension must be decreased, i.e., the new tension should be \(T' = \frac{T}{x}\).

Write the wave speed formula for new tension

Using the formula \(v = \sqrt{\frac{T}{\mu}}\), we can write the new wave speed formula for the decreased tension as follows: \(\frac{1}{2}v = \sqrt{\frac{T'}{\mu}} = \sqrt{\frac{\frac{T}{x}}{\mu}}\)

Solve for the factor x

Now, we need to solve the equation above for the factor x. First, we can square both sides to get rid of the square root: \((\frac{1}{2}v)^2 = (\sqrt{\frac{\frac{T}{x}}{\mu}})^2\) \(\frac{1}{4}v^2 = \frac{T}{x\mu}\) Next, we use the initial wave speed formula \(v = \sqrt{\frac{T}{\mu}}\) and rearrange it to express \(T\) in terms of \(v\) and \(\mu\): \(T = v^2\mu\) Substitute this expression into the equation above: \(\frac{1}{4}v^2 = \frac{v^2\mu}{x\mu}\) Now, we can cancel \(v^2\) and \(\mu\) from both sides and solve for x: \(x = 4\)

Find the correct answer in the given options

The factor by which we must decrease the tension in the string is 4. Therefore, the correct answer is option (d).

Key concepts

Wave Speed Formula

Understanding the wave speed formula is essential when dealing with the motion of waves on a string. The formula itself is expressed as \( v = \sqrt{\frac{T}{\mu}} \), where \( v \) represents the speed of the wave, \( T \) is the tension in the string, and \( \mu \) is the mass per unit length of the string.

From this fundamental relationship, it's clear that wave speed is directly related to the tension applied: the greater the tension, the faster the wave travels. Conversely, if the tension decreases, the speed of the wave also decreases. This principle helps us understand the impact of tension adjustments on wave speed. By modifying the tension, one can control the speed of waves, which is a practical aspect in various technical and musical applications.

When the exercise asks for the wave speed to be reduced to half its current value, it effectively requires us to alter the tension based on this foundational formula. A solid grasp of this relationship between tension and wave speed allows us to manipulate the conditions of wave propagation to our advantage, or in this case, to meet the specific exercise requirement.

Tension in string

Tension plays a crucial role in the behavior of waves on a string. This is because tension is the force that provides the restoring force necessary for the wave to propagate. In the context of our problem, we're investigating how altering this force affects the wave's speed.

Understanding Tension

Tension can be thought of as a pulling force that keeps the string taut, enabling waves to move along its length. It's an intrinsic part of wave dynamics on a string, and it’s important to realize that tension is a scalar quantity that only has magnitude, and no direction.

When it comes to changing the wave's speed, the exercise prompts us to adjust the tension by a specific factor. Since wave speed on a string is proportional to the square root of the tension, reducing the speed to half its original value implies a significant reduction in tension, specifically to one-fourth. This outcome is the square of the speed reduction factor since we're dealing with the square root relationship.

Wave Mechanics

Wave mechanics is the branch of physics that deals with the study of waves. Waves on a string are a common example of mechanical waves that require a medium (in this case, the string) to travel through.

Wave Behavior on Strings

Several factors affect the mechanics of a wave on a string, including tension, mass per unit length, and the way the wave is generated. For the purpose of the exercise, we're primarily concerned with the effect of tension on the wave's propagation speed.

The wave mechanics on a string are governed by principles found in the wave speed formula. Whenever the tension is altered, the speed of the wave responds accordingly. In this educational context, the manipulation of wave speed by adjusting tension offers a tangible demonstration of wave mechanics principles. The exercise illustrates the direct relationship between tension and speed, emphasizing the predictive nature of physical laws in wave behavior.