A transverse wave on a string is described by the following equation.
y(x, t) = (0.35 m) sin[(1.25 rad/m)x + (92.8 rad/s)t]
Consider the element of the string at x = 0.
(a) What is the time interval between the first two instants when this element has a position of y = 0.303 m?
(b) What distance does the wave travel during this time interval?
To find the time interval and distance traveled, we need to find the values of time at which the element has a position of y = 0.303 m.
(a) To find the time interval between the first two instants when the element has a position of y = 0.303 m, we need to solve the equation for t.
y(x, t) = (0.35 m) sin[(1.25 rad/m)x + (92.8 rad/s)t] = 0.303 m
Let's solve for t:
0.303 m = (0.35 m) sin[(1.25 rad/m)(0) + (92.8 rad/s)t]
0.303 = 0.35 sin(92.8t)
Now we can solve for t by taking the inverse sine of both sides:
sin^(-1)(0.303/0.35) = sin^(-1)(sin(92.8t))
Using a calculator, sin^(-1)(0.303/0.35) ≈ 0.791 radians.
Now we divide this value by the angular frequency to find the time interval:
0.791 rad / (92.8 rad/s) ≈ 0.0085 seconds
So, the time interval between the first two instants when this element has a position of y = 0.303 m is approximately 0.0085 seconds.
(b) To find the distance traveled during this time interval, we need to find the wavelength and multiply it by the number of wavelengths traveled in the given time.
The wavelength can be found by dividing the angular frequency by the wave number:
λ = 2π / (1.25 rad/m) = (2π / 1.25) m ≈ 5.03 m
Now we can find the number of wavelengths traveled in the given time interval by dividing the time interval by the period:
T = 2π / (92.8 rad/s) ≈ 0.0677 s
Number of wavelengths = 0.0085 s / 0.0677 s ≈ 0.125
Finally, we can find the distance traveled by multiplying the number of wavelengths by the wavelength:
Distance = 0.125 * 5.03 m ≈ 0.629 m
So, the wave travels approximately 0.629 meters during this time interval.