(a) Write the equation for a transverse sinusoidal wave with a maximum amplitude of 2.5 cm and an angular frequency of 2.9 rad/s that is moving along the positive x-direction with a wave speed that is 5.00 times as fast as the maximum speed of a point on the string. Assume that at time t=0, the point x=0 is at y=0 and then moves in the -y direction in the next instant of time.
(b) Plot the graph for y(x,t) versus x at t= 0 and t=0.01 s.
(c) For that same function plot a graph of y(x,t) versus t at x=0.
So, I think I have the equation right, I used Asin(wt-kx) as my skeleton, and input the values so my equation is 0.025[-sin(2.9t-0.58x)]
But now I have no idea how to graph this.
I have my two graphs set up #1 has x on the x axis and y(x,t) on the y axis, and for #2, I have y(x,t) on the y axis and t on the x axis. Now what do I do? I'm really really bad at graphing.
To graph the equation y(x, t) = 0.025[-sin(2.9t - 0.58x)], we need to understand how the values of x, t, and y change.
(a) To start, let's understand the equation. The general form of a transverse sinusoidal wave is y(x, t) = A sin(kx - wt), where A is the amplitude, k is the wave number, w is the angular frequency, and t is the time.
In this case, the given values are:
Amplitude (A) = 2.5 cm
Angular frequency (w) = 2.9 rad/s
Wave speed (v) = 5 times the maximum speed of a point on the string
We are also given that at time t = 0, the point x = 0 is at y = 0 and then moves in the -y direction in the next instant of time.
Using the given information, we can determine k and v:
k = w / v = 2.9 rad/s / 5v
v = maximum speed of a point on the string
v = w / k = 2.9 rad/s / (5 * 2.9 rad/s / 5v) = 1v
Now, replacing the values in the general form equation, we get:
y(x, t) = 2.5 cm * sin(1x - 2.9t)
(b) To plot the graph for y(x, t) versus x at t = 0 and t = 0.01 s, we need to fix the value of t and vary x.
For t = 0:
y(x, 0) = 2.5 cm * sin(1x)
This equation simplifies to y(x, 0) = 2.5 cm * sin(x)
Plotting this equation, you can choose a range of x values, plug them into the equation, and plot the corresponding y values.
For t = 0.01 s:
y(x, 0.01) = 2.5 cm * sin(x - 0.029)
This equation accounts for the time delay.
Again, choose a range of x values, plug them into the equation, and plot the corresponding y values.
(c) To plot the graph of y(x, t) versus t at x = 0, we need to fix the value of x and vary t.
For x = 0:
y(0, t) = 2.5 cm * sin(-2.9t)
This equation simplifies to y(0, t) = -2.5 cm * sin(2.9t)
Choose a range of t values, plug them into the equation, and plot the corresponding y values.
Remember to label the axes appropriately and scale them based on the given amplitude and time range.
To graph the equation y(x,t) = 0.025[-sin(2.9t-0.58x)], we can follow these steps:
(a) Plot the graph for y(x,t) versus x at t=0 and t=0.01 s:
Step 1: Choose a range of values for x. Let's say x ranges from -10 cm to 10 cm.
Step 2: Substitute t=0 into the equation to find y(x,t) at t=0:
y(x,0) = 0.025[-sin(0-0.58x)]
y(x,0) = 0.025[-sin(-0.58x)]
y(x,0) = 0.025*sin(0.58x)
Step 3: Calculate y(x,t) for various values of x using the equation obtained in step 2. Plot these values on the graph by putting x on the x-axis and y(x,t) on the y-axis. This will give you the graph at t=0.
Step 4: Substitute t=0.01 s into the equation to find y(x,t) at t=0.01 s.
y(x,0.01) = 0.025[-sin(2.9(0.01)-0.58x)]
y(x,0.01) = 0.025[-sin(0.029-0.58x)]
y(x,0.01) = 0.025*sin(0.58x-0.029)
Step 5: Calculate y(x,t) for various values of x using the equation obtained in step 4. Plot these values on the same graph by putting x on the x-axis and y(x,t) on the y-axis. This will give you the graph at t=0.01 s.
(b) For the same function, plot a graph of y(x,t) versus t at x=0:
Step 1: Choose a range of values for t. Let's say t ranges from 0 s to 0.10 s.
Step 2: Substitute x=0 into the equation to find y(x,t) at x=0:
y(0,t) = 0.025[-sin(2.9t-0.58(0))]
y(0,t) = 0.025[-sin(2.9t)]
Step 3: Calculate y(x,t) for various values of t using the equation obtained in step 2. Plot these values on the graph by putting t on the x-axis and y(x,t) on the y-axis. This will give you the graph at x=0.