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Chapter 10: Problem 1373

The maximum velocity and maximum acceleration of a particle executing S.H.M. are \(1 \mathrm{~m} / \mathrm{s}\) and \(3.14 \mathrm{~m} / \mathrm{s}^{2}\) respectively. The frequency of oscillation for this particle is...... (A) \(0.5 \mathrm{~s}^{-1}\) (B) \(3.14 \mathrm{~s}^{-1}\) (C) \(0.25 \mathrm{~s}^{-1}\) (D) \(2 \mathrm{~s}^{-1}\)

Short Answer

Expert verified
The frequency of oscillation for this particle is \(0.5 \mathrm{s}^{-1}\), which corresponds to option (A).

Step by step solution

01

Write down the given information

The maximum velocity (v_max) is given as \(1 \mathrm{~m/s}\) and the maximum acceleration (a_max) is given as \(3.14 \mathrm{~m/s^2}\).
02

Write down the formula for maximum velocity and maximum acceleration in S.H.M.

In a simple harmonic motion, the maximum velocity (v_max) is given by the formula: \(v_{max} = \omega A\) Where \(\omega\) is the angular frequency and \(A\) is the amplitude of oscillation. Similarly, the maximum acceleration (a_max) is given by the formula: \(a_{max} = \omega^2 A\)
03

Find the relationship between maximum velocity and maximum acceleration

We can find the relationship between maximum velocity and maximum acceleration by dividing the two equations from Step 2. So we have: \(\frac{v_{max}}{a_{max}} = \frac{\omega A}{\omega^2 A}\) Solving for \(\omega\) we get: \(\omega =\frac{v_{max}}{a_{max}}\)
04

Calculate the angular frequency

Now that we have a relationship between maximum velocity and maximum acceleration, we can find the angular frequency \(\omega\) using the given values. So, \(\omega = \frac{1 \mathrm{~m/s}}{3.14 \mathrm{~m/s^2}} = \frac{1}{3.14}\mathrm{s^{-1}}\)
05

Calculate the frequency

Now that we have found the angular frequency, we can find the oscillation frequency (f) using the formula: \(f = \frac{\omega}{2\pi}\) Substituting the value of \(\omega\) into the formula we get: \(f = \frac{\frac{1}{3.14}\mathrm{s^{-1}}}{2\pi} = \frac{1}{2\pi \cdot 3.14} \mathrm{s}^{-1} = 0.5 \mathrm{s}^{-1}\) So, the frequency of oscillation for this particle is \(0.5 \mathrm{s}^{-1}\), which corresponds to option (A).

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