Question

A wave along a string has the following equation \( y = 0.02 \sin [30t – 4.0x] \) m. The speed of the wave is:

• A. 4.0 m/s
• B. 30 m/s
• C. 7.5 m/s
• D. 10 m/s

Hint:

The speed of a wave on a string is the ratio of angular frequency to the wave number.

Answer 1

The Correct Option is C

The given wave equation is \( y = 0.02 \sin [30t - 4.0x] \) m.

To find the speed of the wave, we use the general form of the wave equation: \( y = A \sin ( \omega t - kx ) \), where:

• \( \omega \) is the angular frequency
• \( k \) is the wave number
• The wave speed \( v \) is given by: \( v = \frac{\omega}{k} \)

From the given equation:

• \( \omega = 30 \, \text{rad/s} \)
• \( k = 4.0 \, \text{rad/m} \) 

Substitute these values into the formula for wave speed:

\( v = \frac{30}{4.0} = 7.5 \, \text{m/s} \)

Thus, the speed of the wave is 7.5 m/s.

Answer 2

The general form of the wave equation is: \[ y = A \sin(kx - \omega t) \] where:
- \( A \) is the amplitude,
- \( k = 4.0 \, \text{rad/m} \) is the wave number,
- \( \omega = 30 \, \text{rad/s} \) is the angular frequency. The wave speed \( v \) is given by: \[ v = \frac{\omega}{k} \] Substituting the given values: \[ v = \frac{30}{4.0} = 7.5 \, \text{m/s} \] Thus, the speed of the wave is 7.5 m/s.