/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 109 A particle is subjected to two S... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 109

A particle is subjected to two SHM along \(x\) and \(y\) axis, according to \(x=6 \sin 100 \pi t\) and \(y=8 \cos\) \(\left(100 \pi t-\frac{\pi}{2}\right)\), then motion of particle is (A) Ellipse (B) Circle (C) Straight line (D) None of these

Short Answer

Expert verified
The motion of the particle is an ellipse (option A), as the relationship between x and y is given by the equation \(\frac{x^2}{4^2} + \frac{y^2}{3^2} = 1\), which represents an ellipse with semi-major axis a = 4 and semi-minor axis b = 3, centered at the origin.

Step by step solution

01

Write down the given SHM equations for x and y

We are given two SHM equations for x and y axes, which are as follows: 1. \(x = 6\sin(100\pi t)\) 2. \(y = 8\cos(100\pi t - \frac{\pi}{2})\)
02

Simplify the SHM equation for y

To find the link between both motions, it is helpful to have both equations in terms of sine or cosine. Therefore, we simplify the second equation using the trigonometric identity, \(\sin(\alpha - \frac{\pi}{2}) = \cos(\alpha)\). The simplified SHM equation for y is: \(y = 8 \sin(100\pi t)\)
03

Combine the SHM equations for x and y

Divide the first equation by 6: \(\frac{x}{6} = \sin(100\pi t)\) Divide the second equation by 8: \(\frac{y}{8} = \sin(100\pi t)\) Since both equations are equal to \(\sin(100\pi t)\), we can equate them: \(\frac{x}{6} = \frac{y}{8}\)
04

Find the relationship between x and y

Multiply both sides of the equation by 48 to eliminate the fractions: \(8x = 6y\) We can simplify this further: \(\frac{x^2}{4^2} + \frac{y^2}{3^2} = 1\) The equation above represents an ellipse with semi-major axis a = 4 and semi-minor axis b = 3, centered at the origin. Therefore, the motion of the particle is an ellipse (option A).

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