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!
1. The wire tension can be derived from the wave speed, which is
V = 5.9/0.087 = 67.8 m/s,
and the mass per unit length of wire,
sigma = 0.079/5.9 = 1.338*10^-2 kg/m
Now solve the wave speed equation
V = sqrt(T/sigma)
to get the tension in N
T = V^2*sigma = 61.5 N
2. Use the same formulas with a different wire mass.
Thank you so much!
To determine the tension in the wire, we can use the wave speed equation and the relationship between wave speed, tension, and linear mass density.
The wave speed equation is given by:
v = √(T/μ)
Where:
- v is the wave speed (m/s)
- T is the tension in the wire (N)
- μ is the linear mass density (kg/m)
The linear mass density is calculated by dividing the mass of the wire by its length:
μ = m/L
Where:
- m is the mass of the wire (kg)
- L is the length of the wire (m)
Now, let's calculate the tension in the wire.
1. Calculate linear mass density:
μ = m/L = 0.079 kg / 5.9 m
μ ≈ 0.01339 kg/m
2. Calculate wave speed:
v = L / t = 5.9 m / 0.087 s
v ≈ 67.82 m/s
3. Rearrange the wave speed equation to solve for tension:
v = √(T/μ)
T/μ = v^2
T = μ * v^2
4. Substitute the values:
T = 0.01339 kg/m * (67.82 m/s)^2
T ≈ 57.04 N
Therefore, the tension in the wire is approximately 57.04 Newtons.
To calculate the tension for a different mass (part 2):
1. Calculate the linear mass density:
μ = m/L = 0.089 kg / 5.9 m
μ ≈ 0.01508 kg/m
2. Use the tension formula:
T = μ * v^2
3. Substitute the values:
T = 0.01508 kg/m * (67.82 m/s)^2
T ≈ 72.94 N
So, for an 89 g wire, the tension would be approximately 72.94 Newtons.