A heavy piece of hanging sculpture is suspended by a 90 cm long, 5.0 g steel wire. When the
wind blows hard, the wire hums at its fundamental frequency of 80 Hz. What is the mass of the
sculpture?
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Velocity=(0.9)(80)=72m/s
T=V^2*u
=(72)^2*(0.005kg/0.9m)
=28.8 N
T=W
28.8=(mass sculpture + mass wire)*g
..im not sure where I am going wrong...or if any of this is correct..
velocityalongstring= sqrt(tension/mass/length)
velocityalongstring=sqrt(Mass*g*.9/.005)
but the string is one half wavelength long (fixed at both ends), so
from the wave equation
frequency*wavelenth= velocity
frequency*2*.90=sqrt(Mass*9.8*.9/.005)
put in 80 hz, solve for mass of the weight.
You are on the right track. Let's break down the problem step-by-step:
1. The fundamental frequency of the wire hums is given as 80 Hz.
2. The speed of the wave traveling through the wire can be calculated using the formula v = λf, where v is the velocity, λ is the wavelength, and f is the frequency. In this case, the velocity (v) can be calculated as v = 0.9 m (the length of the wire) multiplied by the frequency (f), which is 80 Hz. So, v = 0.9 m * 80 Hz = 72 m/s.
3. The tension in the wire is given by the formula T = μv^2/λ, where T is the tension, μ is the linear mass density (mass per unit length) of the wire, v is the velocity, and λ is the wavelength. In this case, μ is given as 5.0 g (which is equivalent to 0.005 kg) and v is 72 m/s. The wavelength (λ) can be calculated using the formula λ = v/f, where f is the frequency. So, λ = 72 m/s / 80 Hz = 0.9 m.
4. Substituting the values into the tension formula, we have T = (0.005 kg) * (72 m/s)^2 / 0.9 m = 28.8 N.
5. The tension in the wire is equal to the weight of the sculpture plus the weight of the wire, which can be expressed as T = (mass of the sculpture + mass of the wire) * g, where g is the acceleration due to gravity (9.8 m/s^2).
6. Rearranging the equation, we can solve for the mass of the sculpture: mass of the sculpture = (T - mass of the wire * g) / g.
7. Plugging in the values, we have mass of the sculpture = (28.8 N - 0.005 kg * 9.8 m/s^2) / 9.8 m/s^2.
Now, you can calculate the mass of the sculpture using the given values.
To find the mass of the sculpture, we need to calculate the tension (T) in the wire when it hums at its fundamental frequency and equate it to the weight of the sculpture.
First, let's find the tension in the wire using the formula for the fundamental frequency of a hanging wire:
f = (1/2π) * √(T/m)
Where f is the frequency, T is the tension, and m is the mass per unit length of the wire (5.0 g).
Given that the fundamental frequency is 80 Hz, we can rearrange the equation to solve for T:
T = (4π^2 * m * L) / T
Where L is the length of the wire (90 cm = 0.9 m).
T = (4π^2 * m * 0.9) / T
Substituting the values:
80 = (4π^2 * 0.005 * 0.9) / T
Now, we can calculate the tension (T):
T = (4π^2 * 0.005 * 0.9) / 80
T ≈ 0.0178 N
Next, we can equate this tension (T) to the weight of the sculpture:
T = (mass sculpture + mass wire) * g
Since we know the mass of the wire (5.0 g = 0.005 kg) and the acceleration due to gravity (g ≈ 9.8 m/s^2), we can solve for the mass of the sculpture:
0.0178 = (mass sculpture + 0.005) * 9.8
0.0178 = 9.8 * mass sculpture + 0.049
9.8 * mass sculpture = 0.0178 - 0.049
9.8 * mass sculpture = -0.0312
mass sculpture = (-0.0312) / 9.8
mass sculpture ≈ -0.0032 kg
The result comes out to be negative, which doesn't make sense in this context. Therefore, there might be an error in the calculations or the initial information provided.