Answer:
007. 4. 124.091
008. 9. 0.232679738562091
009. 1. 66.8457608738846
010. 3. 14.2 N
Explanation:
007. Speed of a wave is the product of its wavelength and it frequency.
v = λ f
For a given velocity, the minimum frequency occurs at the maximum wavelength.
For a standing wave, the distance between the nodes (fixed points that don't oscillate) is a multiple of half the wavelength.
L = k/2 λ
The wavelength is a maximum at k=1 (also known as the first harmonic).
L = 1/2 λ
λ = 2L
Substituting and solving for f:
v = 2L f
f = v / (2L)
f = 546 m/s / (2 × 2.2 m)
f = 124.091 Hz
008. The sound travels from the dolphin to the ocean floor, then back to the dolphin. So it travels a total distance of 2 × 178 m = 356 m. At a speed of 1530 m/s, the time it takes for the sound to travel this distance is:
t = d / v
t = 356 m / 1530 m/s
t = 0.232679738562091 s
009. Sound intensity in decibels is:
I(db) = 10 log(I / I₀)
where I is the sound intensity (W/m²) and I₀ is the threshold of hearing.
We know that the sound intensity I is proportional to the number of cars per minute. If we say n is the number of cars per minute, and k is the constant of proportionality, then:
I(db) = 10 log(kn / I₀)
When n = 132, I = 73.
73 = 10 log(132k / I₀)
7.3 = log(132k / I₀)
10^7.3 = 132k / I₀
k / I₀ = (10^7.3) / 132
k / I₀ = 151156.236
So the equation for intensity in decibels is:
I(db) = 10 log(151156.236 n)
When n = 32:
I(db) = 10 log(151156.236 × 32)
I(db) = 66.8457608738846
010. For a vibrating string, the tension is:
T = v² m/L
where v is the speed and m/L is the mass per length of the string.
When v = 18.6, T = 6.43.
6.43 = (18.6)² m/L
m/L = 0.01859
So the equation is:
T = 0.01859 v²
When v = 27.6:
T = 0.01859 (27.6)²
T = 14.2 N
