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. \(2.5\) cm
• B. \(5\) cm
• C. \(20\) cm
• D. \(10\) cm

Hint:

The oscillating speed depends on the magnitude of cos of the phase, so points half a wavelength apart have the same speed at all times.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
For \(Y=\sin(20\pi x+10\pi t)\), the particle velocity is \(\frac{\partial Y}{\partial t}=10\pi\cos(20\pi x+10\pi t)\).
The speed is its magnitude, \(|\cos(\text{phase})|\).

Step 2: Condition for equal speed
\(|\cos\phi|\) repeats when the phase differs by \(\pi\). So the phase difference between the points must be \(\pi\).

Step 3: Calculation
Phase difference \(=20\pi\,\Delta x=\pi\Rightarrow\Delta x=\frac{1}{20}\) m \(=5\) cm.

Step 4: Compare the options
The distance \(5\) cm is option (B).

Final Answer:
The minimum distance is 5 cm. \[ \boxed{\text{(B) }5\ \text{cm}} \]