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Chapter 15: Problem 26

Which one of the following equations does not represent SHM, \(x=\) displacement and \(t=\) time. Parameters \(a, b\) and \(c\) are the constants of motion? (a) \(x=a \sin b t\) (b) \(x=a \cos b t+c\) (c) \(x=a \sin b t+c \cos b t\) (d) \(x=a \sec b t+c \operatorname{cosec} b t\)

Short Answer

Expert verified
Option (d) does not represent SHM.

Step by step solution

01

Understand the Properties of SHM

Simple Harmonic Motion (SHM) is characterized by the displacement equation of the form \(x=a\sin(bt + c)\) or \(x=a\cos(bt + c)\), where the sine or cosine function represents oscillation about a mean position. The key feature of SHM is that it involves sinusoidal functions such as sine or cosine.
02

Analyze Option (a)

The equation \(x = a \sin bt\) is a typical SHM equation because it is a pure sinusoidal function (sine in this case). Thus, it represents SHM.
03

Analyze Option (b)

The equation \(x = a \cos bt + c\) can still represent SHM despite the constant term \(c\), because it shifts the function vertically but the core oscillation form remains a cosine function, which is in the SHM form.
04

Analyze Option (c)

The equation \(x = a \sin bt + c \cos bt\) can be rewritten as a single harmonic equivalent using the trigonometric identity for weighted sine and cosine. Therefore, it still maintains the sinusoidal form and represents SHM.
05

Analyze Option (d)

The equation \(x = a \sec bt + c \operatorname{cosec} bt\) does not represent SHM. The functions secant (\(\sec\)) and cosecant (\(\operatorname{cosec}\)) are not sinusoidal functions; instead, they have vertical asymptotes and do not have the oscillatory nature present in sine and cosine functions.
06

Identify the Non-SHM Equation

Based on the analysis, option (d) \(x = a \sec bt + c \operatorname{cosec} bt\) does not represent SHM because it uses non-sinusoidal functions (secant and cosecant) that do not describe simple oscillation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

SHM equations
The heart of Simple Harmonic Motion (SHM) lies in its unique equations that describe oscillations perfectly. SHM is basically oscillatory motion best explained through sinusoidal functions such as sine and cosine.
These functions ensure a continuous and repeating path that an object follows, back and forth around an equilibrium position.
Typically, the general equation for SHM is expressed as:
  • \( x = a \sin(b t + c) \)
  • \( x = a \cos(b t + c) \)
where:
  • \( x \) represents the displacement of the object from its equilibrium point,
  • \( a \) is the amplitude or maximum displacement,
  • \( b \) relates to the angular frequency, dictating how fast the oscillations occur,
  • \( c \) is the phase angle, which indicates the starting position of the object.
These equations capture the essence of SHM by always plotting a smooth, sinusoidal curve that characterizes this type of motion.
Sinusoidal functions
Sinusoidal functions are the backbone of describing oscillatory behavior in physics, especially in Simple Harmonic Motion (SHM).
They include the sine function \( \sin \) and the cosine function \( \cos \), both of which have distinct and predictable patterns.
What makes sinusoidal functions essential in SHM are their properties:
  • **Periodicity:** They repeat values at regular intervals, which means after each complete cycle, the motion repeats itself.
  • **Amplitude:** This is the height of the wave on the graph, representing the maximum extent of oscillation from the mean position.
  • **Frequency and Wavelength:** These are critical for understanding how often the motion occurs and how spread out each cycle is.
In SHM, the sine and cosine functions model how the displacement of an object changes over time, perfectly depicting the nature of smooth, repetitive motions that define oscillations.
Oscillation
Oscillation in the context of Simple Harmonic Motion (SHM) refers to the regular and repeating movement of an object about an equilibrium position.
This movement is often influenced by restoring forces like gravity or tension that try to bring the object back to its original position.
  • **Equilibrium position:** The central point where forces balance and no net force is acting on the object.
  • **Amplitude:** This represents the furthest point from equilibrium, indicating the energy present in the motion.
  • **Period and Frequency:** The period is the time taken for one complete cycle of oscillation, while frequency is the number of oscillations per unit time.
  • **Damping (Not always present):** Real-life oscillations may slow down and stop due to external forces like friction acting against the motion.
Oscillation is a phenomenon seen in pendulums, springs, and even in electrical circuits, and SHM provides a simplified model to analyze such behaviors. It provides a way to understand how objects move and react under cyclical forces and conditions.

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Most popular questions from this chapter

The time period of a mass suspended from a spring is \(5 \mathrm{~s}\). The spring is cut into four equal parts and the same mass is now suspended from one of its parts. The period is now (a) \(5 \mathrm{~s}\) (b) \(2.5 \mathrm{~s}\) (c) \(1.25 \mathrm{~s}\) (d) \(\frac{1}{16} \mathrm{~s}\)

A bottle weighing \(220 \mathrm{~g}\) and area of cross-section \(50 \mathrm{~cm}^{2}\) and height \(4 \mathrm{~cm}\) oscillates on the surface of water in vertical position. Its frequency of oscillation is (a) \(1.5 \mathrm{~Hz}\) (b) \(2.5 \mathrm{~Hz}\) (c) \(3.5 \mathrm{~Hz}\) (d) \(4.5 \mathrm{~Hz}\)

A particle in SHM is described by the displacement function \(x(t)=A \cos (\omega t+\phi), \omega=2 \pi / T\). If the initial ( \(t=0\) ) position of the particle is \(1 \mathrm{~cm}\), its initial velocity is \(\pi \mathrm{cm} \mathrm{s}^{-1}\) and its angular frequency is \(\pi \mathrm{s}^{-1}\), then the amplitude of its motion is (a) \(\pi \mathrm{cm}\) (b) \(2 \mathrm{~cm}\) \(\begin{array}{ll}\text { (c) } \sqrt{2} \mathrm{~cm} & \text { (d) } 1 \mathrm{~cm}\end{array}\)

A mass of \(2.0 \mathrm{~kg}\) is put on a flat pan attached to a vertical spring fixed on the ground as shown in the figure. The mass of the spring and the pan is negligible. When pressed slightly and released the mass executes a simple harmonic motion. The spring constant is \(200 \mathrm{Nm}^{-1}\). What should be the minimum amplitude of the motion, so that the mass gets detached from the pan? (Take \(g=10 \mathrm{~ms}^{-2}\) ) (a) \(8.0 \mathrm{~cm}\) (b) \(10.0 \mathrm{~cm}\) (c) Any value less than \(12.0 \mathrm{~cm}\) (d) \(4.0 \mathrm{~cm}\)

A particle is vibrating in a simple harmonic motion with and amplitude of \(4 \mathrm{~cm}\). At what displacement from the equilibrium position is its energy half potential and half kinetic? (a) \(1 \mathrm{~cm}\) \(\begin{array}{ll}\text { (b) } \sqrt{2} \mathrm{~cm} & \text { (c) } 3 \mathrm{~cm}\end{array}\) (d) \(2 \sqrt{2} \mathrm{~cm}\)
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