To determine the angular frequency of the wave motion given by the equation: \[ y = 0.25 \times 10^{-3} \sin(500t - 0.025x) \] we need to analyze the wave equation in the context of wave properties. ### Step 1: Identify the General Form of the Wave Equation The general form of a sinusoidal wave traveling in the positive x-direction can be expressed as: \[ y(x, t) = A \sin(\omega t - kx) \] where: - \( A \) is the amplitude of the wave, - \( \omega \) is the angular frequency (in radians per second), - \( k \) is the wave number (in radians per meter), - \( t \) is time (in seconds), - \( x \) is the position (in meters). ### Step 2: Extract the Angular Frequency from the Given Equation From the given wave equation: \[ y = 0.25 \times 10^{-3} \sin(500t - 0.025x) \] we can see that the term multiplying \( t \) inside the sine function is \( 500 \). This term corresponds to the angular frequency \( \omega \). ### Step 3: Identify the Angular Frequency Thus, we have: \[ \omega = 500 \, \text{rad/s} \] ### Step 4: Compare with the Options Now, let's compare this value with the provided options: - A. \( 0.25 \times 10^{-3} \, \text{rad/s} \) - B. \( 0.25 \times 10^{-1} \, \text{rad/s} \) - C. \( 5.00 \times 10^{2} \, \text{rad/s} \) - D. \( 2.50 \times 10^{3} \, \text{rad/s} \) The correct answer is **C. \( 5.00 \times 10^{2} \, \text{rad/s} \)**, which is equivalent to \( 500 \, \text{rad/s} \). ### Step 5: Explanation of Incorrect Options - **Option A: \( 0.25 \times 10^{-3} \, \text{rad/s} \)** This value is much too small. It represents \( 0.00025 \, \text{rad/s} \), which is not consistent with the angular frequency derived from the wave equation. - **Option B: \( 0.25 \times 10^{-1} \, \text{rad/s} \)** This value is \( 0.025 \, \text{rad/s} \), which is also significantly lower than the calculated angular frequency of \( 500 \, \text{rad/s} \). - **Option D: \( 2.50 \times 10^{3} \, \text{rad/s} \)** This value is \( 2500 \, \text{rad/s} \), which is higher than the calculated angular frequency. It does not match the coefficient of \( t \) in the sine function. ### Summary of Key Points - The angular frequency \( \omega \) is derived from the coefficient of \( t \) in the wave equation. - The correct angular frequency for the given wave equation is \( 500 \, \text{rad/s} \), which corresponds to option C. - Understanding the structure of the wave equation is crucial for identifying wave properties like angular frequency. ### Revision Summary - The wave equation is of the form \( y = A \sin(\omega t - kx) \). - The angular frequency \( \omega \) is the coefficient of \( t \) in the sine function. - For the given wave equation, \( \omega = 500 \, \text{rad/s} \). - The correct answer is option C: \( 5.00 \times 10^{2} \, \text{rad/s} \).