A string with a mass per length of 2.00 g/m is stretched with a force of 150 N between two points that are 0.500 m apart. The fundamental frequency of the stretched string is in tune with the fundamental frequency of an organ pipe filled with air at 20°C and open at both ends. The speed of sound in air at 0°C is 331 m/s. What is the length of the organ pipe?

To solve this problem, we need to find the frequency of the string first, then use that to find the length of the organ pipe.

1. Find the frequency of the string: The frequency (f) of a vibrating string can be found using the formula: f = (1/2L) * sqrt(F/u), where L is the length of the string, F is the force applied, and u is the mass per unit length.

Substituting the given values: f = (1/(2*0.500m)) * sqrt(150N / 2.00g/m) First, convert the mass per unit length from g/m to kg/m to match the SI unit of force (Newton). So, u = 2.00g/m = 0.002 kg/m. f = (1/1m) * sqrt(150N / 0.002 kg/m) f = sqrt(75000 1/s^2) f = 273.86 Hz

2. Find the speed of sound in air at 20°C: The speed of sound in air increases by 0.6 m/s for each degree Celsius above 0°C. So, the speed of sound at 20°C is: v = 331 m/s + 20°C * 0.6 m/s/°C = 343 m/s.

3. Find the length of the organ pipe: The fundamental frequency of an open pipe is given by the formula: f = v / 2L, where v is the speed of sound and L is the length of the pipe. We can rearrange this to solve for L: L = v / 2f.

Substituting the values we found: L = 343 m/s / (2 * 273.86 Hz) L = 0.626 m

So, the length of the organ pipe is approximately 0.626 m.