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Chapter 10: Problem 1422

The equation for displacement of a particle at time \(t\) is given by the equation \(\mathrm{y}=3 \cos 2 \mathrm{t}+4 \sin 2 \mathrm{t}\). The periodic time of oscillation is \(\ldots \ldots \ldots \ldots\) (A) \(2 \mathrm{~s}\) (B) \(\pi \mathrm{s}\) (C) \((\pi / 2) \mathrm{s}\) (D) \(2 \pi \mathrm{s}\)

Short Answer

Expert verified
The periodic time of oscillation for the given displacement equation is \(\pi\) seconds.

Step by step solution

01

Identify Angular Frequency

Let's first write the given equation: \[y = 3 \cos(2t) + 4 \sin(2t)\] Notice that in both terms of the equation, the argument of the cosine and sine functions is 2t, which means they oscillate at the same frequency. This is a very crucial observation because it tells us the angular frequency involved in the given equation. The angular frequency is represented by the coefficient of t, which in this case is 2.
02

Find the Period of Oscillation

Now that we have identified the angular frequency as 2, we can use it to find the period of oscillation. The formula to convert angular frequency (ω) and the period (T) is given by: \[\omega = \frac{2 \pi}{T}\] In this problem, we have \(\omega = 2\). So, we can write the equation as: \[2 = \frac{2 \pi}{T}\]
03

Solve for T

Now, let's solve for T: \[T = \frac{2 \pi}{2}\] \[T = \pi\] The periodic time of oscillation is \(\pi\) seconds.
04

Determine the Correct Answer

Now that we have calculated the period to be \(\pi\) seconds, we can compare it to the given options. The correct answer is: (B) \(\pi s\)

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Most popular questions from this chapter

A tuning fork of frequency \(480 \mathrm{~Hz}\) produces 10 beats/s when sounded with a vibrating sonometer string. What must have been the frequency of the string if a slight increase in tension produces fewer beats per second than before? (A) \(480 \mathrm{~Hz}\) (B) \(490 \mathrm{~Hz}\) (C) \(460 \mathrm{~Hz}\) (D) \(470 \mathrm{~Hz}\)

A spring is attached to the center of a frictionless horizontal turn table and at the other end a body of mass \(2 \mathrm{~kg}\) is attached. The length of the spring is \(35 \mathrm{~cm}\). Now when the turn table is rotated with an angular speed of \(10 \mathrm{rad} \mathrm{s}^{-1}\), the length of the spring becomes \(40 \mathrm{~cm}\) then the force constant of the spring is $\ldots \ldots \mathrm{N} / \mathrm{m}$. (A) \(1.2 \times 10^{3}\) (B) \(1.6 \times 10^{3}\) (C) \(2.2 \times 10^{3}\) (D) \(2.6 \times 10^{3}\)

If the kinetic energy of a particle executing S.H.M. is given by \(\mathrm{K}=\mathrm{K}_{0} \cos ^{2} \omega \mathrm{t}\), then the displacement of the particle is given by \(\ldots \ldots\) (A) $\left\\{\mathrm{K}_{0} / \mathrm{m} \omega^{2}\right\\} \sin \omega \mathrm{t}$ (B) $\left\\{\left(2 \mathrm{~K}_{0}\right) /\left(\mathrm{m} \omega^{2}\right)\right\\}^{1 / 2} \sin \omega t$ (C) \(\left\\{2 \omega^{2} / \mathrm{mK}_{0}\right\\} \sin \omega t\) (D) $\left\\{2 \mathrm{~K}_{0} / \mathrm{m} \omega\right\\}^{1 / 2} \sin \omega t$

For the following questions, statement as well as the reason(s) are given. Each questions has four options. Select the correct option. (a) Statement \(-1\) is true, statement \(-2\) is true; statement \(-2\) is the correct explanation of statement \(-1\) (b) Statement \(-1\) is true, statement \(-2\) is true but statement \(-2\) is not the correct explanation of statement \(-1\) (c) Statement \(-1\) is true, statement \(-2\) is false (d) Statement \(-1\) is false, statement \(-2\) is true (A) a (B) \(b\) (C) \(\mathrm{c}\) (D) \(\mathrm{d}\) Statement \(-1:\) Periodic time of a simple pendulum is independent of the mass of the bob. Statement \(-2:\) The restoring force does not depend on the mass of the bob. (A) a (B) \(b\) (C) \(\mathrm{c}\) (D) \(\mathrm{d}\)

As shown in figure, two light springs having force constants \(\mathrm{k}_{1}=1.8 \mathrm{~N} \mathrm{~m}^{-1}\) and $\mathrm{k}_{2}=3.2 \mathrm{~N} \mathrm{~m}^{-1}$ and a block having mass \(\mathrm{m}=200 \mathrm{~g}\) are placed on a frictionless horizontal surface. One end of both springs are attached to rigid supports. The distance between the free ends of the spring is \(60 \mathrm{~cm}\) and the block is moving in this gap with a speed \(\mathrm{v}=120 \mathrm{~cm} \mathrm{~s}^{-1}\).What will be the periodic time of the block, between the two springs? (A) \(1+(5 \pi / 6) \mathrm{s}\) (B) \(1+(7 \pi / 6) \mathrm{s}\) (C) \(1+(5 \pi / 12) \mathrm{s}\) (D) \(1+(7 \pi / 12) \mathrm{s}\)
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