A string with mass per unit length 0.0065
kg/m is used to suspend a 30 kg mass over a
pulley. A transverse wave with wavelength 40
cm and amplitude 2.5 mm moves along the
string. Determine the maximum speed of a
particle on the string.
y=A*sin(wt)
A=.0025m
f=wavespeed/wavelength
wavespeed=sqrt(Tension/(m/L))
= sqrt(30*9.8/(.0065)) m/s
wavelength=.4m
figure then frequency f.
w=2PI*f
y=Asin(wt)
y'=Aw cosWt which is particle velocity, so at cos wt=1
max partical velocity= .0025*2PI*f
To determine the maximum speed of a particle on the string, we need to analyze the properties of the wave and the relationship between wave speed and maximum particle velocity.
First, let's calculate the wave speed (v) using the wavelength (λ) and the frequency (f). The wave speed is given by the equation:
v = λf
Given that the wavelength (λ) is 40 cm, we need to convert it to meters by dividing it by 100:
λ = 40 cm / 100 = 0.4 m
Next, we need to find the frequency (f) of the wave. The frequency can be determined using the wave equation:
f = v / λ
Since we have the wave speed (v), we can substitute the known values to find the frequency:
f = v / λ = v / 0.4 m
Now, let's calculate the wave frequency using the given values.
Next, we need to determine the maximum particle velocity (vmax) on the string. The maximum particle velocity is given by the equation:
vmax = Aω
Where A is the amplitude of the wave and ω is the angular frequency. The angular frequency can be calculated using the wave frequency (f) and the formula:
ω = 2πf
Now, we can substitute the values into the equation to find the maximum particle velocity (vmax).
Finally, to determine the maximum speed of a particle on the string, we need to consider that the maximum particle velocity (vmax) occurs when the particle is at its maximum displacement from the equilibrium position. This displacement is equal to the amplitude (A) of the wave.
Therefore, the maximum speed of a particle on the string is equal to the maximum particle velocity (vmax). Thus, we can conclude that the maximum speed of a particle on the string is equal to the maximum particle velocity we calculated earlier.