1 answer
(click or scroll down)
Explain Bot To find the resultant oscillation, we need to add the displacements from the two individual harmonic motions.
Given:
x1 = 8sin(2πt) (cm)
x2 = 4sin(2πt + π/2) (cm)
To add the displacements, we can add the two equations together:
x = x1 + x2
= 8sin(2πt) + 4sin(2πt + π/2)
We can simplify this further by using the trigonometric identity: sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
x = 8sin(2πt) + 4sin(2πt)cos(π/2) + 4cos(2πt)sin(π/2)
= 8sin(2πt) + 4cos(π/2)sin(2πt) + 4cos(2πt)
Now, recall that sin(π/2) = 1 and cos(π/2) = 0, so we can simplify further:
x = 8sin(2πt) + 4sin(2πt) + 4cos(2πt)
= 12sin(2πt) + 4cos(2πt)
This is the equation for the resultant oscillation.
To show it graphically, we can plot the displacement (x) on the y-axis against time (t) on the x-axis. We will need to determine the range for t to generate a reasonable graph.
The period (T) of both SHMs is given by T = 2π/w, where w is the angular frequency. In this case, w = 2π, so T = 2π/(2π) = 1 second.
We can choose a reasonable time range, for example, t = 0 to 2 seconds. Let's plot this on a graph:
- Set up the x-axis for time (t) in seconds.
- Set up the y-axis for displacement (x) in cm.
- Plot the equation x = 12sin(2πt) + 4cos(2πt) for t = 0 to t = 2 seconds.
After plotting the equation, you will see the resultant oscillation graphically.