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Chapter 9: Problem 152

A book with many printing errors contains four different expressions for the displacement \(y\) of a particle executing SHM. Which of the following expressions are wrong? (A) \(y=A \sin \left(\frac{2 \pi t}{T}\right)\) (B) \(y=A \sin v t\) (C) \(y=\frac{A}{T} \sin \left(\frac{t}{A}\right)\) (D) \(y=\frac{A}{\sqrt{2}}(\sin \omega t+\cos \omega t)\)

Short Answer

Expert verified
The incorrect expression for the displacement of a particle executing SHM is (C) \(y=\frac{A}{T} \sin \left(\frac{t}{A}\right)\).

Step by step solution

01

Identify the general equation for SHM

We know that the general equation for simple harmonic motion (SHM) is given by: \(y = A \sin(\omega t + \phi)\) Here, \(A\): amplitude \(\omega\): angular frequency \(\phi\): phase constant
02

Compare expression (A) with the general equation

We need to compare the following expression \(y=A \sin \left(\frac{2 \pi t}{T}\right)\) with the general equation \(y = A \sin(\omega t + \phi)\). In this case, the angular frequency is given by \(\omega = \frac{2 \pi}{T}\), which is correct. There is no phase constant given, which implies it to be zero. Thus, this expression matches the general equation for SHM.
03

Compare expression (B) with the general equation

We need to compare the following expression \(y = A \sin vt\) with the general equation \(y = A \sin(\omega t + \phi)\). Here, \(v\) is used instead of \(\omega\). This may seem incorrect, but we can't conclude that the expression is wrong unless we have more information about the relationship between \(v\) and \(\omega\).
04

Compare expression (C) with the general equation

We need to compare the following expression \(y = \frac{A}{T} \sin \left(\frac{t}{A}\right)\) with the general equation \(y = A \sin(\omega t + \phi)\). In this case, the amplitude \(A\) is multiplied by \(\frac{1}{T}\), which is incorrect. The angular frequency in the expression has a different form from the correct one. Thus, this expression is wrong.
05

Compare expression (D) with the general equation

We need to compare the following expression \(y = \frac{A}{\sqrt{2}}(\sin \omega t + \cos \omega t)\) with the general equation \(y = A \sin(\omega t + \phi)\). This expression can be rewritten as: \(y = A \sin(\omega t + \phi)\) Where \(\phi\) is such that \(\frac{A}{\sqrt{2}}(\sin \omega t + \cos \omega t)\) is equivalent to \(A \sin(\omega t + \phi)\). Although this expression may look different, it is a valid expression for SHM. In conclusion, the incorrect expression is (C).

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Most popular questions from this chapter

A pipe of length \(85 \mathrm{~cm}\) is closed from one end. Find the number of possible natural oscillations of air column in the pipe whose frequencies lie below \(1250 \mathrm{~Hz}\). The velocity of sound in air is \(340 \mathrm{~m} / \mathrm{s}\) (A) 12 . (B) 8 (C) 6 (D) 4

A \(3.6 \mathrm{~m}\) long vertical pipe is filled completely with a liquid. A small hole is drilled at the base of the pipe due to which liquids starts leaking out. This pipe resonates with a tuning fork. The first two resonances occur when height of water column is \(3.22 \mathrm{~m}\) and \(2.34 \mathrm{~m}\), respectively. The area of cross-section of pipe is (A) \(25 \pi \mathrm{cm}^{2}\) (B) \(100 \pi \mathrm{cm}^{2}\) (C) \(200 \pi \mathrm{cm}^{2}\) (D) \(400 \pi \mathrm{cm}^{2}\)

A particle moves on the \(x\)-axis as per the equation \(x=x_{0} \sin ^{2} \omega t .\) The motion is simple harmonic (A) With amplitude \(x_{0}\) (B) With amplitude \(2 x_{0}\) (C) With time period \(\frac{2 \pi}{\omega}\) (D) With time period \(\frac{\pi}{\omega}\)

Equation of a plane progressive wave is given by \(y=0.6 \sin 2 \pi\left(t-\frac{x}{2}\right)\). On reflection from a denser medium, its amplitude becomes \(\frac{2}{3}\) of the amplitude of the incident wave. The equation of the reflected wave is (A) \(y=0.6 \sin 2 \pi\left(t+\frac{x}{2}\right)\) (B) \(y=-0.4 \sin 2 \pi\left(t+\frac{x}{2}\right)\) (C) \(y=0.4 \sin 2 \pi\left(t+\frac{x}{2}\right)\) (D) \(y=-0.4 \sin 2 \pi\left(t-\frac{x}{2}\right)\)

A stretched sonometer wire is in unison with a tuning fork. When the length of the wire is increased by \(2 \%\), the number of beats heard per second is 5 . Then the frequency of the fork is (A) \(245 \mathrm{~Hz}\) (B) \(250 \mathrm{~Hz}\) (C) \(255 \mathrm{~Hz}\) (D) \(260 \mathrm{~Hz}\)
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