A 0.300 g wire is stretched between two points 90.0 cm apart. If the tension in the wire is 600 N, find the wire's first, second, and third harmonics.

To find the first, second, and third harmonics of a wire stretched between two points with a given tension, we can use the formula for the frequency of a stretched wire:

f = (1/2L) * √(T/μ)

Where:
f = frequency
L = length of the wire
T = tension in the wire
μ = linear mass density of the wire (mass per unit length)

First, we need to find the linear mass density μ using the given information. The linear mass density is defined as the mass per unit length of the wire, which can be calculated using the formula:

μ = m/L

Where:
m = mass of the wire
L = length of the wire

Given that the mass of the wire is 0.300 g and the length of the wire is 90.0 cm, we can calculate the linear mass density:

μ = (0.300 g) / (90.0 cm)

Converting the length to meters and the mass to kilograms:

μ = (0.300 g) / (0.900 m)

μ = 0.00333 kg/m

Now, we can use this value of linear mass density and the tension T = 600 N to calculate the frequency of the wire for the first, second, and third harmonics using the formula mentioned earlier. The first harmonic has n = 1, the second harmonic has n = 2, and the third harmonic has n = 3.

Let's calculate the frequencies step-by-step:

First Harmonic (n = 1):
f1 = (1/2L) * √(T/μ)
= (1/2 * 0.900 m) * √(600 N / 0.00333 kg/m)
= 0.556 Hz

Second Harmonic (n = 2):
f2 = (1/2L) * √(2T/μ) [Using f2 = 2 * f1 for harmonics]
= (1/2 * 0.900 m) * √(2 * 600 N / 0.00333 kg/m)
= 1.11 Hz

Third Harmonic (n = 3):
f3 = (1/2L) * √(3T/μ) [Using f3 = 3 * f1 for harmonics]
= (1/2 * 0.900 m) * √(3 * 600 N / 0.00333 kg/m)
= 1.67 Hz

Therefore, the wire's first, second, and third harmonics have frequencies of 0.556 Hz, 1.11 Hz, and 1.67 Hz, respectively.

To find the wire's first, second, and third harmonics, we need to use the equation for the frequency of a wave on a stretched string:

f = (1 / 2L) * √(T / μ)

Where:
f is the frequency of the wave
L is the length of the wire
T is the tension in the wire
μ (mu) is the linear mass density of the wire

First, let's find the linear mass density of the wire. Linear mass density is defined as the mass per unit length of the wire. We can find it by dividing the mass of the wire by its length:

μ = mass / length

Given that the mass of the wire is 0.300 g and the length is 90.0 cm, we can convert the mass to kg and the length to meters to get:

μ = 0.300 g / 90.0 cm = 0.003 kg / 0.90 m = 0.0033 kg/m

Now, we can use the equation for frequency to find the first, second, and third harmonics:

For the first harmonic (n = 1), the frequency is:

f1 = (1 / 2L) * √(T / μ)
= (1 / 2 * 0.90 m) * √(600 N / 0.0033 kg/m)
≈ 199 Hz

For the second harmonic (n = 2), the frequency is:

f2 = 2 * f1
= 2 * 199 Hz
= 398 Hz

For the third harmonic (n = 3), the frequency is:

f3 = 3 * f1
= 3 * 199 Hz
= 597 Hz

Therefore, the wire's first, second, and third harmonics have frequencies of approximately 199 Hz, 398 Hz, and 597 Hz, respectively.