A 5.9 m wire with a mass of 79 g is attached to the mast of a sailboat. If the wire is given a "thunk" at one end, it takes 0.087 s for the resulting wave to reach the other end.
1. What is the tension in the wire?
2. Calculate the tension for a 89-g wire.
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I don't know how to begin this problem, because I don't know how to go about tension in regards to waves. I only ever used tension in the T=m*g format before. And I'm sure I could figure out part 2 if I knew how to do part 1. Thank you for any help!
To find the tension in the wire, you can use the wave speed formula:
Wave speed = frequency x wavelength
The frequency (f) is the reciprocal of the time it takes for the wave to travel the length of the wire (frequency = 1 / time). The wavelength (λ) is twice the length of the wire (λ = 2L).
1. To find the tension for a 79 g wire:
Step 1: Convert the length from meters to centimeters.
5.9 m = 590 cm
Step 2: Find the frequency:
frequency = 1 / time = 1 / 0.087 s = 11.49 Hz
Step 3: Find the wavelength:
wavelength = 2L = 2 x 590 cm = 1180 cm
Step 4: Calculate the wave speed:
wave speed = frequency x wavelength
= 11.49 Hz x 1180 cm
Step 5: Convert the wave speed from cm/s to m/s:
wave speed = (11.49 Hz x 1180 cm) / 100 cm/s
≈ 135.62 m/s
Step 6: Use the wave speed to find the tension using the formula:
wave speed = sqrt(tension / (linear density))
where linear density = mass / length
rearranging the formula gives:
tension = wave speed^2 x linear density
Step 7: Convert the mass from grams to kilograms:
mass = 79 g = 0.079 kg
Step 8: Calculate the linear density:
linear density = mass / length = 0.079 kg / 5.9 m
Step 9: Calculate the tension:
tension = (135.62 m/s)^2 x (0.079 kg / 5.9 m)
2. To find the tension for an 89 g wire, follow the same steps as above but substitute the mass of the wire with 89 g (0.089 kg) in Step 8 and recalculate the tension using the equation in Step 9.
Remember to perform the calculations properly using a calculator.
To solve this problem, you can use the formula for the speed of a wave traveling along a string:
v = √(T/μ)
where:
- v is the wave speed
- T is the tension in the string
- μ is the linear density of the string (mass per unit length)
Step 1: Calculate the wave speed
The wave speed can be calculated using the formula:
v = d / t
where:
- v is the wave speed
- d is the distance traveled by the wave
- t is the time taken for the wave to travel that distance
In this case, the distance traveled by the wave is the length of the wire, which is given as 5.9 m. The time taken is 0.087 s.
Plugging in these values, we have:
v = 5.9 m / 0.087 s
v ≈ 67.816 m/s (rounded to three decimal places)
Step 2: Calculate the linear density
The linear density (μ) is the mass per unit length of the wire. It can be calculated using the formula:
μ = m / L
where:
- μ is the linear density
- m is the mass of the wire
- L is the length of the wire
In this case, the mass of the wire is given as 79 g (0.079 kg) and the length is 5.9 m.
μ = 0.079 kg / 5.9 m
μ ≈ 0.01339 kg/m (rounded to five decimal places)
Step 3: Calculate the tension in the wire
Plugging the values of the wave speed and linear density into the formula for the wave speed, we get:
v = √(T/μ)
(67.816 m/s)^2 = T / 0.01339 kg/m
Simplifying the equation, we have:
T = (67.816 m/s)^2 * 0.01339 kg/m
T ≈ 61.241 N (rounded to three decimal places)
So, the tension in the wire is approximately 61.241 N.
To solve part 2 of the problem, you can follow the same steps but plug in the mass of the wire as 89 g (0.089 kg) in Step 2. Calculate the linear density (μ) and then use it to calculate the tension (T) using the equation T = v^2 * μ.