Strings A and B have the same length and the same tension, but the mass of String B is 2.25 times the mass of string A. Both are vibrating at the same frequency. Which pair (if any) of the diagrams below correctly illustrates possible standing waves on strings A and B? 

If none are possible, choose "none". Select one

A has one half wave, B is a full wave
A is a full wave, B has 1 and a half waves
A has half a wave, B has one and a half waves, 
A has a wave, B has half a wave
None of the illustrations are correct.

To determine which configuration could correctly illustrate standing waves on strings A and B, let's analyze the relationship between the properties of the two strings and their waveforms.

### Given information:
1. **Length and Tension**: Both strings have the same length and the same tension.
2. **Mass**: String B has 2.25 times the mass of String A.
3. **Frequency**: Both strings vibrate at the same frequency.

### Understanding the Wave Speed Relationship:
The wave speed \( v \) on a string is given by the formula:
\[
v = \sqrt{\frac{T}{\mu}}
\]
where:
- \( T \) is the tension in the string (same for both strings),
- \( \mu \) is the linear mass density of the string (mass per unit length).

Since String B has 2.25 times the mass of String A, it has 2.25 times the linear mass density as well. Let \( \mu_A \) be the mass density of String A, and \( \mu_B = 2.25 \mu_A \) for String B.

Substituting into the wave speed formula, we find that:
\[
v_B = \sqrt{\frac{T}{\mu_B}} = \sqrt{\frac{T}{2.25 \mu_A}} = \frac{v_A}{\sqrt{2.25}} = \frac{v_A}{1.5}
\]
This shows that the wave speed on String B is slower than on String A by a factor of 1.5.

### Relating Wave Speed to Wavelength:
Since both strings vibrate at the same frequency \( f \), and wave speed \( v \) is related to frequency and wavelength \( \lambda \) by \( v = f \lambda \), we find:
\[
\lambda_B = \frac{v_B}{f} = \frac{v_A}{1.5 f} = \frac{\lambda_A}{1.5}
\]
This implies that the wavelength on String B is shorter than that on String A by a factor of 1.5.

### Evaluating the Options:
1. **Option 1: A has one half wave, B has a full wave**
   - Here, the wavelength on String B would be half that of String A, implying \( \lambda_B = \frac{\lambda_A}{2} \).
   - This is inconsistent with our calculation (\( \lambda_B = \frac{\lambda_A}{1.5} \)).

2. **Option 2: A is a full wave, B has 1 and a half waves**
   - Here, the wavelength on String B is two-thirds that of String A (\( \lambda_B = \frac{\lambda_A}{1.5} \)), which matches our calculation.
   - This option is **possible**.

3. **Option 3: A has half a wave, B has one and a half waves**
   - Here, the wavelength on String B would be one-third that of String A, implying \( \lambda_B = \frac{\lambda_A}{3} \).
   - This does not match our calculation.

4. **Option 4: A has a wave, B has half a wave**
   - This would imply that the wavelength on String B is double that of String A, which does not match our calculation.

5. **Option 5: None of the illustrations are correct**
   - Since Option 2 is valid, this choice is incorrect.

### Conclusion:
The correct answer is:
**Option 2: A is a full wave, B has 1 and a half waves**

Would you like more details or have any questions?

---

### Related Questions:
1. How does changing the frequency affect the wavelengths on strings A and B?
2. What happens to the wave speed if the tension in both strings is doubled?
3. If the mass of String B were increased to 4 times that of String A, how would the wavelength on B compare to that on A?
4. Why does increasing the mass of a string reduce the wave speed?
5. How would the standing wave patterns change if String A had a higher frequency than String B?

**Tip:** Remember that wave speed in a string depends on both its tension and its linear mass density, making heavier strings vibrate with shorter wavelengths at the same frequency.

Strings A and B have the same length and the same tension, but the mass of String B is 2.25 times the mass of string A. Both are vibrating at the same frequency. Which pair (if any) of the diagrams below correctly illustrates possible standing waves on strings A and B? If none are possible, choose 'none'. Options: A has one half wave, B is a full wave; A is a full wave, B has 1 and a half waves; A has half a wave, B has one and a half waves; A has a wave, B has half a wave; None of the illustrations are correct.

Explore how the mass and tension of strings A and B influence the formation of standing waves. Learn to calculate the wavelength relationship for two strings with the same tension but different masses. This problem involves key wave mechanics concepts and is suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation