The slope of a transversely vibrating string at any point on it is numerically equal to

Options

A. The ratio of the particle speed at that point to the wave speed in the string
B. The ratio of the wave speed in the string to the particle speed at that point
C. One
D. Zero

Answer

To find the numerical value of the slope of a transversely vibrating string at any point on it, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Motion of the String**:
- A transversely vibrating string means that the particles of the string move perpendicular to the direction of the wave propagation.

2. **Assume the Wave Equation**:
- We can represent the displacement of the string as a function of position \( x \) and time \( t \) using the wave equation:
\[ y(x, t) = A \sin(\omega t - kx) \]
- Here, \( A \) is the amplitude, \( \omega \) is the angular frequency, and \( k \) is the wave number.

3. **Calculate the Slope**:
- The slope of the string at any point is given by the partial derivative of \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}(A \sin(\omega t - kx)) \]
- Using the chain rule, we differentiate:
\[ \frac{dy}{dx} = A \cos(\omega t - kx) \cdot (-k) = -Ak \cos(\omega t - kx) \]

4. **Calculate the Velocity of the Particles**:
- The velocity of the particles of the string is given by the partial derivative of \( y \) with respect to \( t \):
\[ \frac{dy}{dt} = \frac{d}{dt}(A \sin(\omega t - kx)) = A \omega \cos(\omega t - kx) \]

5. **Relate Slope to Particle and Wave Velocity**:
- The ratio of the slope to the velocity of the particle can be expressed as:
\[ \frac{\frac{dy}{dx}}{\frac{dy}{dt}} = \frac{-Ak \cos(\omega t - kx)}{A \omega \cos(\omega t - kx)} \]
- Simplifying this gives:
\[ \frac{dy}{dx} = -\frac{k}{\omega} \cdot \frac{dy}{dt} \]
- Since \( \frac{v}{\omega} = \frac{1}{k} \) (where \( v \) is the wave speed), we can express this as:
\[ \frac{dy}{dx} = -\frac{1}{v} \cdot \frac{dy}{dt} \]

6. **Final Expression**:
- The magnitude of the slope is therefore given by:
\[ |\text{slope}| = \frac{|\text{velocity of particle}|}{|\text{velocity of wave}|} \]

### Conclusion:
The slope of a transversely vibrating string at any point is numerically equal to the ratio of the velocity of the particle at that point to the velocity of the wave in the string.