Answer:
Explanation:
The formula holding the relationship between wavelength and frequency of a wave can be expressed as:
[tex]\lambda = \dfrac{v}{f}[/tex]
where;
[tex]\lambda =[/tex] wavelength
v = speed
f = frequency
Given that:
v = 192 m/s and f = 240Hz
Then;
[tex]\lambda = \dfrac{192 \ m/s}{240 \ Hz}[/tex]
[tex]\lambda = 0.800 \ m[/tex]
Now, to estimate the respective amplitude of the string, we need to approach it by using the concept of wave equation which is:
y = A sin kx
here;
A = amplitude of the standing wave
k = wave number
x = maximum displacement
y = distance from center
recall that:
[tex]k = \dfrac{2 \pi}{\lambda}[/tex]
∴
[tex]y = A sin \dfrac{2 \pi}{\lambda }x[/tex]
Now;
for A = 0.400 cm ; [tex]\lambda[/tex] = 0.800 m ; k = 40 cm
Then;
[tex]y =(0.400 \ cm ) \ sin \dfrac{2 \pi}{0.800 m }\times (40 \ cm \times \dfrac{10^{-2} \ m}{1 \ cm } )[/tex]
y = 0.400 sin π
y = 0 cm
At distance 40 cm; the amplitude = 0 cm
Thus, it is a node.
For k = 20cm
Then:
[tex]y =(0.400 \ cm ) \ sin \dfrac{2 \pi}{0.800 m }\times (20 \ cm \times \dfrac{10^{-2} \ m}{1 \ cm } )[/tex]
y = 0.400 sin π/2
y = 0.400 cm
At distance 20 cm; the amplitude = 0.400 cm
Thus, it is antinode.
For k = 10cm
Then:
[tex]y =(0.400 \ cm ) \ sin \dfrac{2 \pi}{0.800 m }\times (10 \ cm \times \dfrac{10^{-2} \ m}{1 \ cm } )[/tex]
4y = 0.400 sin π/2
y = 0.283 cm
At distance 10 cm; the amplitude = 0.283 cm
b)
The required time taken to go through the displacement( i.e. from largest upward to downward) is the time required to cover half of the wavelength.
This is expressed as:
[tex]T = \dfrac{1}{2} \times \dfrac{1}{f}[/tex]
[tex]T= \dfrac{1}{2} \times \dfrac{1}{240 \ Hz}[/tex]
T = 0.00208
T = 2.08 × 10⁻³ s
