The equation of a wave traveling along the positive x-direction is given by; y = 0.25 x 10 ^ -3 sin (500t…

1. Analyze the Question

You are given the wave equation $y = 0.25 \times 10^{-3} \sin(500t - 0.025x)$ and asked to find the angular velocity (also known as angular frequency, denoted by $\omega$).

2. Explain the Concept

The standard equation for a progressive wave traveling in the positive x-direction is:

Where:

• $A$ is the amplitude.
• $\omega$ is the angular velocity (angular frequency) in $\text{rads}^{-1}$.
• $t$ is the time.
• $k$ is the wave number.
• $x$ is the displacement.

By comparing the given equation to the standard form, you can identify the value of $\omega$ by looking at the coefficient of $t$.

3. Show the Solution

Given equation:

Standard equation:

By direct comparison:

• $\omega = 500$

To express $500$ in standard scientific notation as shown in the options:

Therefore, the angular velocity $\omega = 5.00 \times 10^2 \text{ rads}^{-1}$.

4. Identify the Answer

Comparing our result to the options provided: A. $0.25 \times 10^{-3} \text{ rads}^{-1}$ B. $0.25 \times 10^{-1} \text{ rads}^{-1}$ C. $5.00 \times 10^2 \text{ rads}^{-1}$ D. $2.50 \times 10^3 \text{ rads}^{-1}$

The correct option is C.

5. Tip

When you see a wave equation, don't let the numbers intimidate you! Always write the standard equation ($y = A \sin(\omega t - kx)$) directly above or below the given equation. This makes it easy to "map" the values: the number attached to $t$ is always your angular velocity ($\omega$), and the number attached to $x$ is always your wave number ($k$).