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?
The sine term must be 0.8657 if y(x,t) = 0.303. That happens at 1.0466 radians and at pi - 1.0466, or 2.0950 radians.
At x = 0, that means 92.8 t = 1.0466 or 2.0950
t = 11.3 msec or 22.6 msec
(a) Well, since we're dealing with a sinusoidal wave here, we need to figure out when the element of the string reaches a position of y = 0.303 m for the first and second time. Let me take out my imaginary wave calculator... Okay, got it. So, the first instant is at t = 0.0082 s, and the second instant is at t = 0.0252 s.
Now, we just need to find the time interval between these two instants. Let's do some math-y stuff here. The time interval is given by the equation Δt = t2 - t1. Plugging in the values we found, we get Δt = 0.0252 s - 0.0082 s = 0.017 s. Voilà, there's your answer!
(b) To find the distance the wave travels during this time interval, we need to use the wave speed equation, v = λf. But, since we don't have the frequency, we'll have to find it using the wave equation. Hang on while I dust off my math skills again... Alright, f = ω/2π. Plugging in the given angular frequency ω = 92.8 rad/s, we get f = 92.8/(2π) ≈ 14.8 Hz.
Now, the velocity of the wave is given as v = 1.25 m/ rad * 14.8 Hz ≈ 18.5 m/s. To find the distance traveled, we just multiply the velocity by the time interval: d = v * Δt ≈ 18.5 m/s * 0.017 s ≈ 0.3145 m. So the wave travels a distance of approximately 0.3145 meters during this time interval. Hope that made sense, otherwise, I'll have to go back to clown school!
To find the time interval when the element has a position of y = 0.303 m, we can set the equation y(x, t) = 0.303 m and solve for t.
0.303 m = (0.35 m)sin[(1.25 rad/m)x + (92.8 rad/s)t]
To solve for t, we can isolate the angle inside the sine function:
sin[(1.25 rad/m)x + (92.8 rad/s)t] = 0.303 m / 0.35 m
Taking the inverse sine (sin^(-1)) of both sides gives us:
(1.25 rad/m)x + (92.8 rad/s)t = sin^(-1)(0.303/0.35)
Now we can solve for t:
t = [sin^(-1)(0.303/0.35) - (1.25 rad/m)x] / (92.8 rad/s)
Now that we have the value of t, we can calculate the time interval between the first two instants when the element has a position of y = 0.303 m.
(a) The time interval between the first two instants when the element has a position of y = 0.303 m is given by the difference in time at those two instants. So, the time interval is:
Δt = t₂ - t₁
where t₂ and t₁ are the two instants when y = 0.303 m. We need more information to calculate the values of t₂ and t₁.
(b) Similarly, to calculate the distance the wave travels during this time interval, we need to know the speed of the wave on the string (v). Then, the distance traveled is given by:
Distance = v × time interval (Δt)
Again, we need more information to calculate the distance traveled.
To find the time interval between the first two instants when the element has a position of y = 0.303 m, we need to set y(x, t) equal to 0.303 m and solve for t.
The equation y(x, t) = (0.35 m) sin[(1.25 rad/m)x + (92.8 rad/s)t] represents a transverse wave on a string. The general form of a transverse wave is given by y(x, t) = A sin(kx - ωt), where A is the amplitude of the wave, k is the wave number (2π divided by the wavelength), x is the position along the string, ω is the angular frequency (2π times the frequency), and t is the time.
Comparing the given equation to the general form, we can determine the values of A, k, ω, and the desired y:
A = 0.35 m (amplitude)
k = 1.25 rad/m (wave number)
x = 0 (position, specifically x = 0)
y = 0.303 m (desired position)
Substituting these values into the equation, we have:
0.303 m = (0.35 m) sin[(1.25 rad/m)(0) + (92.8 rad/s)t]
0.303 m = (0.35 m) sin(92.8 rad/s t)
To solve for t, we can isolate t by dividing both sides by (0.35 m) and taking the inverse sine of both sides:
sin⁻¹(0.303 m / 0.35 m) = sin⁻¹(sin(92.8 rad/s t))
Now, we can use a calculator to find the inverse sine of the fraction on the left-hand side to obtain the value of the angle:
sin⁻¹(0.303 m / 0.35 m) = 0.866 radians
Next, we can substitute this angle back into the equation to solve for t:
0.866 radians = 92.8 rad/s t
Dividing both sides by 92.8 rad/s, we find:
t = 0.00934 seconds
So, the time interval between the first two instants when the element has a position of y = 0.303 m is approximately 0.00934 seconds.
To calculate the distance the wave travels during this time interval, we need to find the speed of the wave. The speed of a transverse wave on a string is given by the formula v = ω/k, where v is the speed, ω is the angular frequency, and k is the wave number.
The angular frequency ω is given as 92.8 rad/s, and the wave number k is given as 1.25 rad/m. Substituting these values into the formula, we have:
v = (92.8 rad/s) / (1.25 rad/m)
v ≈ 74.24 m/s
Now, we can calculate the distance traveled using the formula distance = speed × time. Substituting the values we've found, we have:
distance = (74.24 m/s) × (0.00934 seconds)
distance ≈ 0.6919 meters
Therefore, the wave travels approximately 0.6919 meters during this time interval.