A harmonic wave on a string moves along the x-axis according to:
î(x,t) = 5.0 (cm) sin[ (2.5m-1) x - (63s-1) t ]
At x = 1.0 m when t = 2.2 s the speed of the oscillations on the string is closest to ...
25.2 m/s
4.55 m/s
3.15 m/s
2.27 m/s
1.67 m/s
its definitely not 25.2 or 3.15
then what is is? how do you solve this?? im so confused :(
To find the speed of the oscillations on the string, we need to determine the velocity of a point on the string at x = 1.0 m and t = 2.2 s.
The wave equation given can be written as:
î(x,t) = 5.0 cm sin[(2.5 m^(-1))x - (63 s^(-1))t]
To find the velocity, we need to take the derivative of the equation with respect to time (t). Taking the derivative, we get:
v = ∂î/∂t = 5.0 cm (63 s^(-1)) cos[(2.5 m^(-1))x - (63 s^(-1))t]
Now, substitute the values x = 1.0 m and t = 2.2 s into the equation to find the speed of the oscillations:
v = 5.0 cm (63 s^(-1)) cos[(2.5 m^(-1))(1.0 m) - (63 s^(-1))(2.2 s)]
v = 315 cm/s cos(1.5π - 138.6π)
v = 315 cm/s cos(-137.1π)
Since cos(-θ) = cos(θ), we can simplify the equation to:
v = 315 cm/s cos(137.1π)
Now, calculate the cosine of 137.1π. Using a calculator or computer, we find that cos(137.1π) is approximately -0.9798.
v ≈ 315 cm/s * (-0.9798)
v ≈ -308.07 cm/s
Taking the magnitude of the velocity to find the speed:
Speed = |v| ≈ |-308.07 cm/s| = 308.07 cm/s
Converting cm/s to m/s:
Speed ≈ 308.07 cm/s * (1 m/100 cm) ≈ 3.0807 m/s
Therefore, the speed of the oscillations on the string, when x = 1.0 m and t = 2.2 s, is closest to 3.15 m/s.