Construct a 5th mode standing wave on a string that has an unknown mass hanging from one end. The distance between fixed ends of the string is 0.9 meters. If the wave speed on the 1.3 meter string is 216 meters per seconds and the string has a mass of 3 grams, what is the value for the unknown mass and frequency? Instead of solving for the answer explain how to solve this problem.
To solve this problem, we need to use the principles of wave mechanics and the equation for the wave speed on a string. Here are the steps to find the unknown mass and frequency:
1. Determine the wave speed: Given that the wave speed on the string is 216 m/s, this represents the speed at which the waves travel through the string. This value will be crucial in solving the problem.
2. Find the tension in the string: The wave speed on a string is related to the tension in the string (T) and the mass per unit length (μ) by the equation v = √(T/μ). Since the mass per unit length is not given directly, we need to use the information given in the problem to find it.
3. Calculate the mass per unit length: The mass per unit length (μ) of the string can be calculated by dividing the total mass of the string by its length. In this case, the mass of the string is given as 3 grams, and the length of the string is 1.3 meters.
4. Determine the tension in the string: Using the equation v = √(T/μ), we can rearrange it to solve for tension (T). Since we know the wave speed (v) and have calculated the mass per unit length (μ), we can plug these values into the equation and solve for tension.
5. Find the unknown mass: Now that we have the tension in the string, we can move on to finding the unknown mass (m). The tension in the string is equal to the weight of the unknown mass hanging from one end of the string. We can use the formula T = mg, where g is the acceleration due to gravity (approximately 9.8 m/s^2), to find the unknown mass.
6. Calculate the frequency: Once we have the tension and the unknown mass, we can use the equation for the frequency of a standing wave on a string, f = (n/2L)√(T/μ), to find the frequency. In this case, the mode of the standing wave is given as the 5th mode (n = 5), and the distance between fixed ends of the string is 0.9 meters (L = 0.9 m).
By following these steps, you will be able to find the values for the unknown mass and frequency in the given scenario.