1) The wave speed is 15.2 m/s
2) The tension in the slinky is 33.2 N
3) The average speed of a piece of the slinky during one pulse is 0.475 m/s
4) The wavelength is 34.5 m
Explanation:
1)
The motion of a wave pulse along the slinky is a uniform motion, therefore its speed is given by the equation for uniform motion:
[tex]v=\frac{L}{t}[/tex]
where
L is the length covered
t is the time elapsed
For the wave in this problem, we have:
L = 6.2 m is the length of the slinky
t = 0.408 s is the time taken for a pulse to travel across the length os the slinky
Substituting, we find the wave speed
[tex]v=\frac{6.2}{0.408}=15.2 m/s[/tex]
2)
The speed of a wave on a slinky can be found with the same expression for the wave speed along a string:
[tex]v=\sqrt{\frac{T}{m/L}}[/tex]
where
T is the tension in the slinky
m is the mass of the slinky
v is the wave speed
L is the length
In this problem, we have:
m = 0.89 kg is the mass of the slinky
L = 6.2 m is the length
Therefore, we can re-arrange the equation to find the tension in the slinky, T:
[tex]T=v^2 (\frac{m}{L})=(15.2)^2 (\frac{0.89}{6.2})=33.2 N[/tex]
3)
The average speed of a piece of the slinky as a complete wave pulse passes is the total displacement done by a piece of slinky during one period, which is 4 times the amplitude, divided by the time taken for one complete oscillation, the period:
[tex]v_{avg} = \frac{4A}{T}[/tex]
where
A is the amplitude
T is the period
Here we have:
A = 0.27 m is the amplitude of the wave
The period is the reciprocal of the frequency, therefore
[tex]T=\frac{1}{f}[/tex]
where f = 0.44 Hz is the frequency of this wave. Substituting and solving, we find
[tex]v_{avg} = \frac{4A}{1/f}=4Af=4(0.27)(0.44)=0.475 m/s[/tex]
4)
The wavelength of the wave pulse can be found by using the wave equation:
[tex]v=f\lambda[/tex]
where
v is the wave speed
f is the frequency
[tex]\lambda[/tex] is the wavelength
For the pulse in this problem, we have
v = 15.2 m/s
f = 0.44 Hz
Substituting, we find the wavelength:
[tex]\lambda=\frac{v}{f}=\frac{15.2}{0.44}=34.5 m[/tex]
Learn more about waves:
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