A transverse wave on a string is described by the following equation.
y(x, t) = (0.35 m) sin[(1.25 rad/m)x + (81.2 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 = 0.247 m?
(in miliseconds)
(b) What distance does the wave travel during this time interval?
(in meters)
To answer these questions, we need to use the equation given and extract the necessary information.
(a) To find the time interval between the first two instants when the element has a position of y = 0.247 m, we need to determine the values of t at which y = 0.247 m.
The equation provided is y(x, t) = (0.35 m) sin[(1.25 rad/m)x + (81.2 rad/s)t]. Comparing this equation with the given value, we can equate the two expressions:
0.247 m = (0.35 m) sin[(1.25 rad/m)x + (81.2 rad/s)t].
Taking the inverse sine of both sides, we have:
sin^(-1)(0.247/0.35) = (1.25 rad/m)x + (81.2 rad/s)t.
Now we need to solve for t. Rearranging the equation, we get:
(81.2 rad/s)t = sin^-1(0.247/0.35) - (1.25 rad/m)x.
Dividing both sides by 81.2 rad/s, we obtain:
t = (1/81.2 rad/s) * (sin^-1(0.247/0.35) - (1.25 rad/m)x).
Using a calculator, plug in the values to get the result in seconds. To convert the seconds to milliseconds, multiply by 1000.
(b) To find the distance the wave travels during this time interval, we need to find the displacement of the element between the two instants when y = 0.247 m.
The equation y(x, t) = (0.35 m) sin[(1.25 rad/m)x + (81.2 rad/s)t] gives us the y-coordinate of the element at any given x and t.
To find the distance traveled, we need to subtract the initial position at t = 0 from the final position at t = calculated time from part (a).
Therefore, plug in the calculated time into the equation y(x, t) and subtract the initial position y(x=0, t=0). The result will give you the displacement in meters.