For a particle executing simple harmonic motion, the maximum speed and acceleration are 10 cm/s and 50 cm/s^2 , respectively. Determine the position(s) of the particle when its speed is 8 cm/s .

Correct answer: D. ± 1.2 cm from the mean position Given v_ = Aω = 10 cm/s and a_ = Aω^2 = 50 cm/s^2 . Dividing the two expressions, we get: ω = (Aω^2)/(Aω) = (50)/(10) = 5 rad/s The amplitude A is: A = (10)/(ω) = (10)/(5) = 2 cm The velocity v of a particle in SHM at a distance x from the mean position is given by: v = ω A^2 - x^2 Substituting v = 8 cm/s , ω = 5 rad/s , and A = 2 cm : 8 = 5 2^2 - x^2 Squaring both sides: 64 = 25(4 - x^2) 25x^2 = 100 - 64 = 36 x^2 = (36)/(25) = 1.44 x = ± 1.2 cm

Concepts Of Physics MCQ Edition [Volume 1]

Exam: Concepts Of Physics MCQ Edition [Volume 1]
Subject: Physics
Chapter: Simple Harmonic Motion

Concepts Of Physics MCQ Edition [Volume 1] · Physics · Simple Harmonic Motion

For a particle executing simple harmonic motion, the maximum speed...

For a particle executing simple harmonic motion, the maximum speed and acceleration are

10 \text{ cm/s}

and

50 \text{ cm/s}^2

, respectively. Determine the position(s) of the particle when its speed is

8 \text{ cm/s}

Options

A $\pm 1.6 \text{ cm}$ from the mean position
B $\pm 1.5 \text{ cm}$ from the mean position
C $\pm 0.8 \text{ cm}$ from the mean position
D $\pm 1.2 \text{ cm}$ from the mean position

Correct answer

D. $\pm 1.2 \text{ cm}$ from the mean position

Step-by-step solution

Given

v_{\text\max} = A\omega = 10 \text{ cm/s}

and

a_{\text\max} = A\omega^2 = 50 \text{ cm/s}^2

.

Dividing the two expressions, we get:

\omega = \dfrac{A\omega^2}{A\omega} = \dfrac{50}{10} = 5 \text{ rad/s}

The amplitude

A

is:

A = \dfrac{10}{\omega} = \dfrac{10}{5} = 2 \text{ cm}

The velocity

v

of a particle in SHM at a distance

x

from the mean position is given by:

v = \omega \sqrt{A^2 - x^2}

Substituting

v = 8 \text{ cm/s}

\omega = 5 \text{ rad/s}

, and

A = 2 \text{ cm}

8 = 5 \sqrt{2^2 - x^2}

Squaring both sides:

64 = 25(4 - x^2)

25x^2 = 100 - 64 = 36

x^2 = \dfrac{36}{25} = 1.44

x = \pm 1.2 \text{ cm}

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