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Chapter 9: Problem 49

A particle executing SHM has a maximum speed of \(30 \mathrm{~cm} / \mathrm{s}\) and maximum acceleration of \(60 \mathrm{~cm} / \mathrm{s}^{2}\). The period of oscillation is (A) \(\pi \mathrm{s}\) (B) \(\frac{\pi}{2} \mathrm{~s}\) (C) \(2 \pi \mathrm{s}\) (D) \(\frac{\pi}{t} \mathrm{~s}\)

Short Answer

Expert verified
The period of oscillation for the given SHM is \(蟺 \mathrm{s}\) (option A).

Step by step solution

01

Recall the SHM formulas

In simple harmonic motion, the maximum speed is given by \(v_{max} = 蠅A\), where \(蠅\) is the angular frequency and \(A\) is the amplitude. The maximum acceleration is given by \(a_{max} = 蠅^2 A\).
02

Solve the equations for angular frequency and amplitude

We are given \(v_{max} = 30 \mathrm{cm/s}\) and \(a_{max} = 60 \mathrm{cm/s^2}\). We can set up the following equations using the SHM formulas: \( v_{max} = 蠅A \quad \Rightarrow \quad 30 = 蠅A\) \( a_{max} = 蠅^2 A \quad \Rightarrow \quad 60 = 蠅^2 A\)
03

Divide the two equations

To eliminate A, we can divide the two equations: \(\frac{60}{30} = \frac{蠅^2A}{蠅A}\) Divide by the terms on both sides to get the angular frequency: \(蠅 = 2 \mathrm{rad/s}\)
04

Determine the period

The relationship between angular frequency and period is given by: \(蠅 = \frac{2蟺}{T}\) where T denotes the period. Substitute the value of 蠅 and solve for T: \(2 = \frac{2蟺}{T}\) \(T = \frac{2蟺}{2}\) \(T = 蟺 \mathrm{s}\) So the period of oscillation is \(蟺 \mathrm{s}\), which corresponds to option (A).

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Most popular questions from this chapter

A tuning fork of frequency \(340 \mathrm{~Hz}\) is vibrated just above a cylindrical tube of length \(120 \mathrm{~cm}\). Water is slowly poured in the tube. If the speed of sound is 340 \(\mathrm{m} / \mathrm{s}\), then the minimum height of water required for resonance is (A) \(25 \mathrm{~cm}\) (B) \(45 \mathrm{~cm}\) (C) \(75 \mathrm{~cm}\) (D) \(95 \mathrm{~cm}\)

The total energy of a particle, executing simple harmonic motion is (A) independent of \(x\) (B) \(\propto x^{2}\) (C) \(\propto x\) \((\mathrm{D}) \propto x^{1 / 2}\) where \(x\) is the displacement from the mean position.

A closed organ pipe of length \(L\) vibrating in second overtone, then match the following Column-I (A) Displacement node (B) Displacement anti-node (C) Pressure node (D) Pressure anti-node Column-II 1\. Closed end 2\. Open end 3\. \(4 L / 5\) from closed end 4\. \(L / 5\) from closed end

The track followed for two perpendicular SHMs is a perfect ellipse when \(\left(\delta\right.\)-phase difference, \(A_{1}, A_{2}\) amplitudes) (A) \(\delta=\frac{\pi}{4}, A_{1} \neq A_{2}\) (B) \(\delta=\frac{3 \pi}{4}, A_{1} \neq A_{2}\) (C) \(\delta=\frac{\pi}{2}, A_{1} \neq A_{2}\) (D) \(\delta=\pi, A_{1}=A_{2}\)

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