A 3.00 kHz tone is being produced by a speaker with a diameter of 0.175 m. The air temperature changes from 0 to 29°C. Assuming air to be an ideal gas, find the change in the diffraction angle

My solution

v directly proportional to sqrt T

v = sqrt 302 =17.382

lambda = v/f = 17.382/3000
=0.005794

Sin theta = 1.22 lambda/ diameter

= 1.22x 0.005794/0.175
=0.009425
theta = Sin^-1 0.009425
= 0.54

answer is wrong

We are given the initial temperature T1 = 0°C and the final temperature T2 = 29°C. The speed of sound in air changes with temperature as follows:

v(T) = v0 * sqrt(T/T0)

where v0 is the speed of sound at a reference temperature T0. We can use the speed of sound at 0°C as our reference, v0 = 331.5 m/s (according to the standard model of the atmosphere).

The initial speed of sound (at 0°C) is v1 = v0 = 331.5 m/s. The final speed of sound (at 29°C) is:

v2 = v0 * sqrt((T2 + 273) / (T1 + 273))
v2 = 331.5 * sqrt((29 + 273) / 273)
v2 ≈ 347.6 m/s

Now we can compute the initial and final wavelengths:

lambda1 = v1 / f = 331.5 / 3000 ≈ 0.1105 m
lambda2 = v2 / f = 347.6 / 3000 ≈ 0.1159 m

The diffraction angle can be found using the formula:

sin(theta) = 1.22 * lambda / diameter

First, let's compute the initial angle:

sin(theta1) = 1.22 * 0.1105 / 0.175
sin(theta1) ≈ 0.7709
theta1 = arcsin(0.7709) ≈ 50.4°

Now let's compute the final angle:

sin(theta2) = 1.22 * 0.1159 / 0.175
sin(theta2) ≈ 0.8079
theta2 = arcsin(0.8079) ≈ 53.6°

Finally, the change in the diffraction angle is:

delta_theta = theta2 - theta1 ≈ 53.6 - 50.4 ≈ 3.2°

Your solution seems to be missing a step. After calculating the value of theta in radians, you need to convert it to degrees (since the question asks for the change in diffraction angle), using the formula:

θ_degrees = θ_radians * (180 / π)

Let's calculate it step by step:

v ∝ √T
v = √302 = 17.382 m/s

λ = v/f = 17.382 / 3000 = 0.005794 m

Sin(θ) = (1.22 * λ) / diameter
Sin(θ) = (1.22 * 0.005794) / 0.175
Sin(θ) = 0.04025

θ_radians = sin^(-1)(0.04025) ≈ 0.0403 rad

θ_degrees = 0.0403 * (180 / π) ≈ 2.3125 degrees

So, the change in the diffraction angle is approximately 2.3125 degrees.

To find the change in the diffraction angle due to the change in air temperature, you need to take into account the speed of sound in air at different temperatures.

The relationship between the speed of sound in air and temperature can be given by the equation:

v = sqrt(gamma * R * T)

Where:
v is the speed of sound in air,
gamma is the heat capacity ratio for air (approximately 1.4),
R is the gas constant for air (approximately 287 J/(kg*K)),
T is the air temperature in Kelvin.

First, convert the temperatures from Celsius to Kelvin. 0°C = 273K, and 29°C = 302K.

So, the initial speed of sound (v1) at 0°C is given by:
v1 = sqrt(1.4 * 287 * 273) = 331.5 m/s

The final speed of sound (v2) at 29°C is given by:
v2 = sqrt(1.4 * 287 * 302) = 343.9 m/s

The change in the speed of sound (delta v) is:
delta v = v2 - v1 = 343.9 - 331.5 = 12.4 m/s

To find the change in the diffraction angle, you need to determine the change in wavelength (delta lambda), which can be calculated using the formula:

delta lambda = (delta v / v1) * lambda

where lambda is the initial wavelength, given by:
lambda = v1 / f = 331.5 / 3000 = 0.1105 m

Plugging in the values:
delta lambda = (12.4 / 331.5) * 0.1105 = 0.0041 m

Now, to find the change in the diffraction angle (delta theta), you can use the formula:

delta theta = arcsin(delta lambda / diameter)

where the diameter is given as 0.175m.

delta theta = arcsin(0.0041 / 0.175) = 0.024 radians

This is the change in the diffraction angle caused by the change in air temperature.