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Chapter 10: Problem 1441

The function \(\sin ^{2}(\omega t)\) represents (A) A SHM with periodic time \(\pi / \omega\) (B) A SHM with a periodic time \(2 \pi / \omega\) (C) A periodic motion with periodic time \(\pi / \omega\) (D) A periodic motion with period \(2 \pi / \omega\)

Short Answer

Expert verified
The given function \(\sin^2(\omega t)\) represents a periodic motion (not a Simple Harmonic Motion) with a period of \(2 \pi / \omega\). Therefore, the correct answer is (D) A periodic motion with period \(2 \pi / \omega\).

Step by step solution

01

Recognize the waveform

The given function is \(\sin^2(\omega t)\), which is a trigonometric function squared. It represents a non-negative, periodic motion. However, it is not a Simple Harmonic Motion (SHM) as its shape is not sinusoidal. So options (A) and (B) can be discarded.
02

Evaluate the period of the given function

To find the period of the motion, we need to find the value of \(t\) for which the function repeats its cycle. Let's look for the period of the motion given by \(\sin^2(\omega t)\): We know that the sine function has a period of \(2\pi\), which means: \[\sin(\omega t) = \sin(\omega t + 2 \pi n)\] where \(n\) is an integer. Simply square both sides to find the period of the given function: \[\sin^2(\omega t) = \sin^2(\omega t + 2 \pi n)\] Thus, \(\sin^2(\omega t)\) has the same period as the sine function, which is \(2 \pi\). To find the period concerning \(t\), we need to divide by the coefficient of \(t\) in the argument, which is \(\omega\): Period of \(\sin^2(\omega t) = \dfrac{2 \pi}{\omega}\)
03

Conclusion

Now we know that the given function \(\sin^2(\omega t)\) describes a periodic motion (not an SHM) with period \(2 \pi / \omega\). So the correct answer is: (D) A periodic motion with period \(2 \pi / \omega\).

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Most popular questions from this chapter

As shown in figure, a spring attached to the ground vertically has a horizontal massless plate with a \(2 \mathrm{~kg}\) mass in it. When the spring (massless) is pressed slightly and released, the \(2 \mathrm{~kg}\) mass, starts executing S.H.M. The force constant of the spring is \(200 \mathrm{Nm}^{-1}\). For what minimum value of amplitude, will the mass loose contact with the plate? (Take \(\left.\mathrm{g}=10 \mathrm{~ms}^{-2}\right)\) (A) \(10.0 \mathrm{~cm}\) (B) \(8.0 \mathrm{~cm}\) (C) \(4.0 \mathrm{~cm}\) (D) For any value less than \(12.0 \mathrm{~cm}\).

If the equation for displacement of two particles executing S.H.M. is given by \(\mathrm{y}_{1}=2 \sin (10 \mathrm{t}+\theta)\) and $\mathrm{y}_{2}=3 \cos 10 \mathrm{t}$ respectively, then the phase difference between the velocity of two particles will be \(\ldots \ldots \ldots\) (A) \(-\theta\) (B) \(\theta\) (C) \(\theta-(\pi / 2)\) (D) \(\theta+(\pi / 2)\).

The distance travelled by a particle performing S.H.M. during time interval equal to its periodic time is \(\ldots \ldots\) (A) A (B) \(2 \mathrm{~A}\) (C) \(4 \mathrm{~A}\) (D) Zero.

When a mass \(M\) is suspended from the free end of a spring, its periodic time is found to be \(\mathrm{T}\). Now, if the spring is divided into two equal parts and the same mass \(\mathrm{M}\) is suspended and oscillated, the periodic time of oscillation is found to be \(\mathrm{T}\) '. Then \(\ldots \ldots \ldots\) (A) \(\mathrm{T}<\mathrm{T}^{\prime}\) (B) \(\mathrm{T}=\mathrm{T}^{\prime}\) (C) \(\mathrm{T}>\mathrm{T}^{\prime}\) (D) Nothing can be said.

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) \(\mathrm{b}\) (C) \(c\) (D) \(\mathrm{d}\) Statement \(-1:\) For a particle executing SHM, the amplitude and phase is decided by its initial position and initial velocity. Statement \(-2:\) In a SHM, the amplitude and phase is dependent on the restoring force. (A) a (B) \(b\) (C) \(\mathrm{c}\) (D) \(\mathrm{d}\)
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