Two waves passing through a region are represented by `y=(1.0cm) sin [(3.14 cm^(-1))x - (157s^(-1) x - (157s^(-1))t]`
and `y = (1.5 cm) sin [(1.57 cm^(-1))x- (314 s^(-1))t].` Find the displacement of the particle at x = 4.5 cm at time t = 5.0 ms.

Explanation

To find the displacement of the particle at \( x = 4.5 \, \text{cm} \) and time \( t = 5.0 \, \text{ms} \) due to the two waves represented by the equations:

1. \( y_1 = (1.0 \, \text{cm}) \sin \left( (3.14 \, \text{cm}^{-1})x - (157 \, \text{s}^{-1})t \right) \)
2. \( y_2 = (1.5 \, \text{cm}) \sin \left( (1.57 \, \text{cm}^{-1})x - (314 \, \text{s}^{-1})t \right) \)

we will follow these steps:

### Step 1: Calculate \( y_1 \)

Substituting \( x = 4.5 \, \text{cm} \) and \( t = 5.0 \, \text{ms} = 5.0 \times 10^{-3} \, \text{s} \) into the equation for \( y_1 \):

\[ y_1 = (1.0 \, \text{cm}) \sin \left( (3.14 \, \text{cm}^{-1})(4.5 \, \text{cm}) - (157 \, \text{s}^{-1})(5.0 \times 10^{-3} \, \text{s}) \right) \]

Calculating the argument of the sine function:

\[ = (3.14 \times 4.5) - (157 \times 0.005) \]
\[ = 14.13 - 0.785 = 13.345 \]

Now, we can find \( y_1 \):

\[ y_1 = (1.0 \, \text{cm}) \sin(13.345) \]

### Step 2: Calculate \( y_2 \)

Now substituting the same values into the equation for \( y_2 \):

\[ y_2 = (1.5 \, \text{cm}) \sin \left( (1.57 \, \text{cm}^{-1})(4.5 \, \text{cm}) - (314 \, \text{s}^{-1})(5.0 \times 10^{-3} \, \text{s}) \right) \]

Calculating the argument of the sine function:

\[ = (1.57 \times 4.5) - (314 \times 0.005) \]
\[ = 7.065 - 1.57 = 5.495 \]

Now, we can find \( y_2 \):

\[ y_2 = (1.5 \, \text{cm}) \sin(5.495) \]

### Step 3: Calculate the net displacement \( y \)

Using the principle of superposition, the net displacement \( y \) is given by:

\[ y = y_1 + y_2 \]

Substituting the values we calculated:

\[ y = (1.0 \, \text{cm}) \sin(13.345) + (1.5 \, \text{cm}) \sin(5.495) \]

### Step 4: Final Calculation

Now we compute the sine values and sum them up:

1. Calculate \( \sin(13.345) \) and \( \sin(5.495) \) using a calculator.
2. Substitute these values back into the equation for \( y \).

### Conclusion

The final value of \( y \) will give us the displacement of the particle at \( x = 4.5 \, \text{cm} \) and \( t = 5.0 \, \text{ms} \).

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