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Chapter 10: Problem 1421

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 motion of the particle is \(\ldots \ldots\) (A) Damped motion (B) Periodic motion (C) Rotational motion (D) S.H.M.

Short Answer

Expert verified
The motion of the particle is (B) Periodic motion, as determined by analyzing the equation \(y = 3\cos2t + 4\sin2t\) and rewriting it as \(y(t) = 5\cos(2t-\tan^{-1}(\frac{4}{3}))\), which represents a periodic cosine function.

Step by step solution

01

Rewrite the displacement equation in terms of sine and cosine functions

Let's rewrite the equation as: \(y(t) = A\cos(2t - \phi)\) Where A is the amplitude and \(\phi\) is the phase angle. To do this, we need to find the values of A and \(\phi\).
02

Calculate A and \(\phi\)

From the given equation \(y = 3\cos2t + 4\sin2t\), we have a set of simultaneous equations: \(A\cos\phi = 3\) \(A\sin\phi = 4\) To find the amplitude, A, take the square root of the sum of the squares: \(A = \sqrt{(3^2+4^2)} = 5\) To find the phase angle \(\phi\), take the inverse tangent of the ratio: \(\phi = \tan^{-1}(\frac{4}{3})\)
03

Rewrite the displacement equation with A and \(\phi\)

Now, we can rewrite the displacement equation with the calculated values of A and \(\phi\): \(y(t) = 5\cos(2t-\tan^{-1}(\frac{4}{3}))\)
04

Identify the motion type

The displacement equation represents a cosine function, which is a periodic function. Therefore, the motion of the particle is periodic. The correct answer is (B) Periodic motion.

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Most popular questions from this chapter

A block having mass \(\mathrm{M}\) is placed on a horizontal frictionless surface. This mass is attached to one end of a spring having force constant \(\mathrm{k}\). The other end of the spring is attached to a rigid wall. This system consisting of spring and mass \(\mathrm{M}\) is executing SHM with amplitude \(\mathrm{A}\) and frequency \(\mathrm{f}\). When the block is passing through the mid-point of its path of motion, a body of mass \(\mathrm{m}\) is placed on top of it, as a result of which its amplitude and frequency changes to \(\mathrm{A}^{\prime}\) and \(\mathrm{f}\). The ratio of amplitudes $\left(\mathrm{A}^{1} / \mathrm{A}\right)=\ldots \ldots \ldots$ (A) \(\sqrt{\\{}(\mathrm{M}+\mathrm{m}) / \mathrm{m}\\}\) (B) \(\sqrt{\\{m} /(\mathrm{M}+\mathrm{m})\\}\) (C) \(\sqrt{\\{} \mathrm{M} /(\mathrm{M}+\mathrm{m})\\}\) (D) \(\sqrt{\\{}(\mathrm{M}+\mathrm{m}) / \mathrm{M}\\}\)

An listener is moving towards a stationary source of sound with a speed (1/4) times the speed of sound. What will be the percentage increase in the frequency of sound heard by the listener? (A) \(20 \%\) (B) \(25 \%\) (C) \(2.5 \%\) (D) \(5 \%\)

Two waves are represented by $\mathrm{y}_{1}=\mathrm{A} \sin \omega \mathrm{t}\( and \)\mathrm{y}_{2}=\mathrm{A}\( cos \)\omega \mathrm{t}$. The phase of the first wave, \(\mathrm{w}\). r. t. to the second wave is (A) more by radian (B) less by \(\pi\) radian (C) more by \(\pi / 2\) (D) less by \(\pi / 2\)

A closed organ pipe has fundamental frequency \(100 \mathrm{~Hz}\). What frequencies will be produced if its other end is also opened? (A) \(200,400,600,800 \ldots \ldots\) (B) \(200,300,400,500 \ldots \ldots\) (C) \(100,300,500,700 \ldots \ldots\) (D) \(100,200,300,400 \ldots \ldots\)

The average values of potential energy and kinetic energy over a cycle for a S.H.O. will be ............ respectively. (A) \(0,(1 / 2) \mathrm{m} \omega^{2} \mathrm{~A}^{2}\) (B) \((1 / 2) \mathrm{m} \omega^{2} \mathrm{~A}^{2}, 0\) (C) $(1 / 2) \mathrm{m} \omega^{2} \mathrm{~A}^{2},(1 / 2) \mathrm{m} \omega^{2} \mathrm{~A}^{2}$ (D) $(1 / 4) \mathrm{m} \omega^{2} \mathrm{~A}^{2},(1 / 4) \mathrm{m} \omega^{2} \mathrm{~A}^{2}$
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