The function y(x, t) = (25.0 cm) cos(ðx - 15ðt), with x in meters and t in seconds, describes a wave on a taut string. What is the transverse speed for a point on the string at an instant when that point has the displacement y = +22.0 cm?
To find the transverse speed for a point on the string at a specific instant when the point has a displacement of y = +22.0 cm, we need to differentiate the equation for y(x, t) with respect to time (t):
1. Find the derivative of y(x, t) with respect to t:
dy/dt = -15ð(25.0 cm) sin(ðx - 15ðt)
2. Substitute y = +22.0 cm into the derivative:
22.0 cm/s = -15ð(25.0 cm) sin(ðx - 15ðt)
3. Solve for the transverse speed by isolating sin(ðx - 15ðt):
sin(ðx - 15ðt) = (22.0 cm/s) / (-15ð(25.0 cm))
Explanation:
In wave motion, the displacement function y(x, t) describes the variation of the transverse displacement of a point on a waveform as a function of the position (x) and time (t).
To find the transverse speed, we take the derivative of the displacement function with respect to time (t). This will give us a function that represents the rate of change of displacement with respect to time, which is the definition of speed.
By substituting the given displacement y = +22.0 cm into the derivative, we can solve for the transverse speed.
Keep in mind that we are assuming a point on the string usually moves in a simple harmonic manner along with the wave, thus the sine function relates the displacement and time.