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 between two instants when the element of the string has a position of y = 0.303 m, we need to find the values of t when y(x,t) = 0.303 m.
Given the equation for the transverse wave on the string:
y(x, t) = (0.35 m) sin[(1.25 rad/m)x + (92.8 rad/s)t]
To find the time interval between the first two instants when y = 0.303 m, we need to solve the equation for t.
Step 1: Set y(x, t) = 0.303 m
0.303 m = (0.35 m) sin[(1.25 rad/m)x + (92.8 rad/s)t]
Step 2: Divide both sides by 0.35 m
0.303 m / 0.35 m = sin[(1.25 rad/m)x + (92.8 rad/s)t]
Step 3: Take the inverse sine of both sides
sin^(-1)(0.303 m / 0.35 m) = (1.25 rad/m)x + (92.8 rad/s)t
Step 4: Solve for t
t = [sin^(-1)(0.303 m / 0.35 m) - (1.25 rad/m)x] / (92.8 rad/s)
Now we have an expression for t in terms of x. To find the time interval between the first two instants when y = 0.303 m, we need to subtract the value of t when x = 0 from the value of t when x = 0. Since we are only interested in the time interval, we can subtract the two values of t.
(a) The time interval between the first two instants when the element has a position of y = 0.303 m is:
t = [sin^(-1)(0.303 m / 0.35 m) - (1.25 rad/m)(0)] / (92.8 rad/s) - [sin^(-1)(0.303 m / 0.35 m) - (1.25 rad/m)(0)] / (92.8 rad/s)
Simplifying this expression will give the time interval in seconds.
To find the distance the wave travels during this time interval, we can use the wave speed.
The wave speed can be determined from the equation: v = λf, where v is the wave speed, λ is the wavelength, and f is the frequency.
From the given equation of the wave: y(x, t) = (0.35 m) sin[(1.25 rad/m)x + (92.8 rad/s)t], we can see that the wave number k = 1.25 rad/m.
The wavelength λ is given by the relation: λ = 2π/k.
Using the equation v = λf, we can find the wave speed v and the frequency f.
(b) The distance the wave travels during the time interval can be calculated using the wave speed v and the time interval obtained in part (a). The formula to calculate the distance traveled by a wave is given by: distance = velocity × time.
Therefore, the distance the wave travels during this time interval is: distance = v × time interval. Substitute the values of v and the time interval to get the distance in meters.