A wave is described by the equation y = (1.0 mm) π ( x 2.0 cm - t 0.01 s ). What are the speeds of the particles at x = 3.0 cm , 5.0 cm and 7.0 cm at t = 0.01 s ?

Correct answer: B. zero for all three positions The given wave equation is y = (1.0 mm) π ( x 2.0 cm - t 0.01 s ). The particle velocity v_p is given by the partial derivative of y with respect to t : v_p = (∂ y)/(∂ t) = ( 1.0 mm) π ((x)/(2.0) - (t)/(0.01)) × (-(π)/(0.01)) Since 1.0 mm = 0.1 cm , we get: v_p = -10π π ((x)/(2.0) - (t)/(0.01)) cm s^ -1 At t = 0.01 s , the term (t)/(0.01) = 1 . Substituting this into the velocity equation: v_p = -10π ((π x)/(2.0) - π) = 10π ((π x)/(2.0)) cm s^ -1 Evaluating the particle speed at the given positions: For x = 3.0 cm , v_p = 10π ((3π)/(2)) = 0 For x = 5.0 cm , v_p = 10π ((5π)/(2)) = 0 For x = 7.0 cm , v_p = 10π ((7π)/(2)) = 0 Thus, the speeds of the particles are zero for all three positions.

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). What are the speeds of the particles at

x = 3.0 \text{ cm}

5.0 \text{ cm}

and

7.0 \text{ cm}

t = 0.01 \text{ s}

Options

A $10\pi \text{ cm s}^{-1}$ for all three positions
Bzero for all three positions
C $5\pi \text{ cm s}^{-1}$ for all three positions
Dzero, $5\pi \text{ cm s}^{-1}$ , and $10\pi \text{ cm s}^{-1}$ respectively

Correct answer

B. zero for all three positions

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

v_p

is given by the partial derivative of

with respect to

:<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>Since

1.0 \text{ mm} = 0.1 \text{ cm}

, we get:<br><br>

v_p = -10\pi \cos \pi \left(\dfrac{x}{2.0} - \dfrac{t}{0.01}\right) \text{ cm s}^{-1}

<br><br>At

t = 0.01 \text{ s}

, the term

\dfrac{t}{0.01} = 1

. Substituting this into the velocity equation:<br><br>

v_p = -10\pi \cos \left(\dfrac{\pi x}{2.0} - \pi\right) = 10\pi \cos \left(\dfrac{\pi x}{2.0}\right) \text{ cm s}^{-1}

<br><br>Evaluating the particle speed at the given positions:<br><br>For

x = 3.0 \text{ cm}

,

v_p = 10\pi \cos \left(\dfrac{3\pi}{2}\right) = 0

<br><br>For

x = 5.0 \text{ cm}

,

v_p = 10\pi \cos \left(\dfrac{5\pi}{2}\right) = 0

<br><br>For

x = 7.0 \text{ cm}

,

v_p = 10\pi \cos \left(\dfrac{7\pi}{2}\right) = 0

<br><br>Thus, the speeds of the particles are zero for all three positions.

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