A wave is described by the equation y = (1.0 mm) π ( x 2.0 cm - t 0.01 s ). Find the speed of the particle at x = 1.0 cm at time t = 0.01 s .

Correct answer: B. zero The given wave equation is: y = (1.0 mm) π ( x 2.0 cm - t 0.01 s ) The particle velocity is given by the partial derivative of displacement with respect to time: v_p = (∂ y)/(∂ t) = ( 1.0 mm) π ((x)/(2.0) - (t)/(0.01)) × (-(π)/(0.01)) Substituting x = 1.0 cm and t = 0.01 s: v_p = -100π π ((1.0)/(2.0) - (0.01)/(0.01)) v_p = -100π ((π)/(2) - π) v_p = -100π (-(π)/(2)) Since (-(π)/(2)) =0 , we get: v_p = 0 Thus, the speed of the particle is zero.

Concepts Of Physics MCQ Edition [Volume 1]

Exam: Concepts Of Physics MCQ Edition [Volume 1]
Subject: Physics
Chapter: Wave Motion and Waves on a String

Concepts Of Physics MCQ Edition [Volume 1] · Physics · Wave Motion and Waves on a String

A wave is described by the equation y = (1.0 mm) π ( x 2.0 cm - t...

A wave is described by the equation

y = (1.0 \text{ mm}) \sin \pi

\left(\dfrac{x}{2.0 \text{ cm}} - \dfrac{t}{0.01 \text{ s}}\right). Find the speed of the particle at

x = 1.0 \text{ cm}

at time

t = 0.01 \text{ s}

Options

A $5\pi \text{ cm s}^{-1}$
Bzero
C $10\pi \text{ cm s}^{-1}$
D $20\pi \text{ cm s}^{-1}$

Correct answer

B. zero

Step-by-step solution

The given wave equation is:

y = (1.0 \text{ mm}) \sin \pi \left(\dfrac{x}{2.0 \text{ cm}} - \dfrac{t}{0.01 \text{ s}}\right)<br><br>The particle velocity is given by the partial derivative of displacement with respect to time:<br><br>

v_p =

\dfrac{\partial y}{\partial t} = (

1.0

\text{ mm}) \cos \pi \left(\dfrac{x}{2.0} - \dfrac{t}{0.01}\right) \times \left(-\dfrac{\pi}{0.01}\right)<br><br>Substituting

x = 1.0

\text{ cm} and

t = 0.01

\text{ s}:<br><br>

v_p =

-100\pi \cos \pi \left(\dfrac{1.0}{2.0} - \dfrac{0.01}{0.01}\right)<br><br>

v_p =

-100\pi \cos

\left(\dfrac{\pi}{2} - \pi\right)

v_p =

-100\pi \cos \left(-\dfrac{\pi}{2}\right)

Since

\cos \left(-\dfrac{\pi}{2}\right) =0

, we get:

v_p = 0

Thus, the speed of the particle is zero.

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