Question

The equation of a wave travelling on a string is $ y = \sin[20\pi x + 10\pi t] $, where x and t are distance and time in SI units. The minimum distance between two points having the same oscillating speed is :

• A. 5.0 cm
• B. 20 cm
• C. 10 cm
• D. 2.5 cm

Hint:

The oscillating speed of points on a sinusoidal wave has the same spatial periodicity as the wave itself (the wavelength). Therefore, the minimum distance between two points having the same oscillating speed (at the same time) is equal to the wavelength of the wave.

Answer 1

The Correct Option is C

Step 1: Identify the Wave Parameters
The given wave equation is: \[ y = \sin(20\pi x + 10\pi t) \] The general form of a traveling wave is: \[ y = \sin(kx + \omega t + \phi) \] Comparing the given equation with the general form:

• Wave number (\( k \)) = \( 20\pi \, \text{rad/m} \)
• Angular frequency (\( \omega \)) = \( 10\pi \, \text{rad/s} \)

Step 2: Determine the Wavelength (\( \lambda \))
The wavelength is related to the wave number by: \[ k = \frac{2\pi}{\lambda} \] \[ \lambda = \frac{2\pi}{k} = \frac{2\pi}{20\pi} = 0.1 \, \text{m} = 10 \, \text{cm} \] \subsection{Step 3: Find the Oscillating Speed (\( v \))} The oscillating speed is the time derivative of the displacement: \[ v = \dv{y}{t} = \dv{t} \sin(20\pi x + 10\pi t) \] \[ v = 10\pi \cos(20\pi x + 10\pi t) \]
Step 4: Condition for Same Oscillating Speed
For two points to have the same oscillating speed at any instant, their phase angles must satisfy: \[ \cos(\theta_1) = \cos(\theta_2) \] This implies: \[ \theta_2 = \theta_1 + 2n\pi \quad \text{or} \quad \theta_2 = -\theta_1 + 2n\pi \] Given \( \theta = 20\pi x + 10\pi t \), the phase difference \( \Delta \theta \) must satisfy: \[ \Delta \theta = 20\pi \Delta x = 2n\pi \quad \text{or} \quad \Delta \theta = 20\pi \Delta x = -2\theta_1 + 2n\pi \] The smallest non-zero distance occurs when: \[ 20\pi \Delta x = \pi \] \[ \Delta x = \frac{\pi}{20\pi} = \frac{1}{20} \, \text{m} = 5 \, \text{cm} \] \subsection{Final Answer} The minimum distance between two points with the same oscillating speed is \(5.0 \, \text{cm}\).