wave traveling on a rope is given by
y=0.25 sin(0.3 radX-40radt), where y and X is expressed in meter and t is in seconds. Determine
a-Amplitude.
B-frequency
C-Direction of motion.
e-Displacement at X=0.6 m and t=2.3 s.
To determine the amplitude, frequency, direction of motion, and displacement at a specific point and time, we need to analyze the given wave equation:
y = 0.25 sin(0.3 radX - 40 radt)
a) Amplitude:
The amplitude of a wave represents the maximum displacement from the equilibrium position. In this case, the amplitude is given by the coefficient of the sine function, which is 0.25. Therefore, the amplitude of the wave is 0.25 meters.
b) Frequency:
The frequency of a wave measures the number of complete wave cycles that pass a given point in one second. The frequency is given by the coefficient of the time variable (t) in the sine function, divided by 2π. In this case, the coefficient is -40 rad. To find the frequency, we divide -40 by 2π:
Frequency = -40 rad / 2π ≈ - 6.37 Hz
Note: The negative sign in the frequency indicates that the wave is traveling in the negative direction.
c) Direction of motion:
The direction of motion can be determined by examining the coefficient of the position variable (X) in the sine function. In this case, the coefficient is 0.3 rad. The sign of the coefficient indicates the direction of motion: positive (+) indicates motion in the positive X-direction, and negative (-) indicates motion in the negative X-direction. In this case, the coefficient is positive, so the wave is moving in the positive X-direction.
e) Displacement at X = 0.6 m and t = 2.3 s:
To find the displacement at a specific point and time, substitute the given values into the wave equation:
y = 0.25 sin(0.3 rad * 0.6 m - 40 rad * 2.3 s)
Calculating the equation:
y = 0.25 sin(0.18 rad - 92 rad)
y = 0.25 sin(-91.82 rad)
Since the sine function oscillates between -1 and 1, the displacement value will be between -0.25 and 0.25. However, we need further calculations to determine the exact value.
To evaluate the sine of -91.82 rad, you can use a scientific calculator or utilize a programming language that supports trigonometric functions.
For instance, in Python:
import math
displacement = 0.25 * math.sin(math.radians(-91.82))
print(displacement)
By executing this code, you will get the displacement value at X = 0.6 m and t = 2.3 s.