If a wave y(x,t)=(5.0 mm) sin(kr +(600 rad/s) `t+phi)` travels along a string, how much time does any given point on the string take to move between displacements y=+2.0 mm and y=-2.0 mm?

Answer

To solve the problem of how much time a given point on the string takes to move between displacements \( y = +2.0 \, \text{mm} \) and \( y = -2.0 \, \text{mm} \), we can follow these steps:

### Step 1: Understand the wave equation
The wave is given by:
\[ y(x, t) = (5.0 \, \text{mm}) \sin(kx + (600 \, \text{rad/s}) t + \phi) \]
Here, \( A = 5.0 \, \text{mm} \), \( \omega = 600 \, \text{rad/s} \), and \( k \) is the wave number.

### Step 2: Set up equations for displacements
To find the times \( t_1 \) and \( t_2 \) when the displacement is \( +2.0 \, \text{mm} \) and \( -2.0 \, \text{mm} \), we set up the following equations:

1. For \( y = +2.0 \, \text{mm} \):
\[ 5.0 \sin(kx + 600 t_1 + \phi) = 2.0 \]
Simplifying gives:
\[ \sin(kx + 600 t_1 + \phi) = \frac{2.0}{5.0} = \frac{2}{5} = 0.4 \]

2. For \( y = -2.0 \, \text{mm} \):
\[ 5.0 \sin(kx + 600 t_2 + \phi) = -2.0 \]
Simplifying gives:
\[ \sin(kx + 600 t_2 + \phi) = -\frac{2.0}{5.0} = -0.4 \]

### Step 3: Use inverse sine to find angles
Now we can find the angles corresponding to these sine values:
1. For \( t_1 \):
\[ kx + 600 t_1 + \phi = \sin^{-1}(0.4) \]
Let \( \theta_1 = \sin^{-1}(0.4) \).

2. For \( t_2 \):
\[ kx + 600 t_2 + \phi = \sin^{-1}(-0.4) \]
Let \( \theta_2 = \sin^{-1}(-0.4) \).

### Step 4: Find the time difference
Now, we can subtract the two equations:
\[ 600 t_1 - 600 t_2 = \theta_1 - \theta_2 \]
This simplifies to:
\[ 600 (t_1 - t_2) = \theta_1 - \theta_2 \]
Thus, we can express the time difference as:
\[ t_1 - t_2 = \frac{\theta_1 - \theta_2}{600} \]

### Step 5: Calculate the angles
Using the properties of sine:
- \( \theta_1 = \sin^{-1}(0.4) \)
- \( \theta_2 = -\sin^{-1}(0.4) \)

So:
\[ \theta_1 - \theta_2 = \sin^{-1}(0.4) - (-\sin^{-1}(0.4)) = 2 \sin^{-1}(0.4) \]

### Step 6: Substitute and calculate time
Substituting back into the equation for time:
\[ t_1 - t_2 = \frac{2 \sin^{-1}(0.4)}{600} \]

### Step 7: Calculate \( \sin^{-1}(0.4) \)
Using a calculator:
\[ \sin^{-1}(0.4) \approx 0.4115 \, \text{radians} \]
Thus:
\[ t_1 - t_2 = \frac{2 \times 0.4115}{600} \approx \frac{0.823}{600} \approx 0.001372 \, \text{s} \approx 1.37 \, \text{ms} \]

### Final Answer
The time taken for any given point on the string to move between displacements \( y = +2.0 \, \text{mm} \) and \( y = -2.0 \, \text{mm} \) is approximately \( 1.37 \, \text{ms} \).

---