/*! 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 49 The position of a particle as a ... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 49

The position of a particle as a function of time is \(\vec{r}=4 \sin 2 \pi t \hat{i}+4 \cos 2 \pi t \hat{j}\) (where \(t\) is time in second). Path of this particle will be (A) an ellipse (B) a hyperbola (C) a circle (D) any other curved path

Short Answer

Expert verified
The particle moves in a circular path

Step by step solution

01

Interpret Function Components

Break down the position function. It is composed of two components, one for x coordinate and one for y coordinate. The x component is \(4 \sin 2 \pi t\) and the y component is \(4 \cos 2 \pi t\). Each component is a trigonometric function (sine or cosine) of input \(2 \pi t\), where t represents time.
02

Analyzing the Functions

At first glance, the functions of sine and cosine have the same amplitude and frequency. This is a common trait of the parametric equations of a circular path, where x and y coordinates oscillate harmonically, maintaining the same frequency and amplitude. We can deduce the motion to be a circular motion already here, yet for clarity, we should investigate more closely.
03

Converting into Cartesian Coordinates

To have a more intuitive understanding of the path, convert the parametric oscillatory motion into a Cartesian equation. For this, it is helpful to remember that \(\sin^2 x + \cos^2 x = 1\). Squaring both the x and y components and adding them together gives us \(16 \sin^2 2 \pi t + 16 \cos^2 2 \pi t = 16(\sin^2 2 \pi t + \cos^2 2 \pi t ) = 16 * 1 = 16\). The resulting equation represents a circle of radius 4 in the Cartesian Plane.
04

Final Result

As the resulting equation represents the equation of a circle with radius 4, we can conclude the path of the particle to be a circle.

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Most popular questions from this chapter

A road is \(5 \mathrm{~m}\) wide. Its radius of curvature is \(20 \sqrt{6} \mathrm{~m}\). The outer edge is above the inner edge by a distance of \(1 \mathrm{~m}\). This road is most suited for a speed \(\left(g=10 \mathrm{~ms}^{-2}\right.\) ) (A) \(10 \mathrm{~ms}^{-1}\) (B) \(10 \sqrt{5} \mathrm{~ms}^{-1}\) (C) \(100 \mathrm{~ms}^{-1}\) (D) \(40 \sqrt{6} \mathrm{~ms}^{-1}\)

A glass wind screen whose inclination with the vertical can be changed is mounted on a car. The car moves horizontally with a speed of \(2 \mathrm{~m} / \mathrm{s}\). If the angle of the wind screen with vertical is \(\theta\) when vertically downward falling raindrops with velocity of \(6 \mathrm{~m} / \mathrm{s}\) strikes the screen perpendicularly. Find \(\tan \theta\).

A projectile is fired with a velocity \(u\) at right angle to a slope, which is inclined at an angle \(\theta\) with the horizontal. The range of the projectile on the incline is (A) \(\frac{2 u^{2} \sin \theta}{g}\) (B) \(\frac{2 u^{2}}{g} \tan \theta \sec \theta\) (C) \(\frac{u^{2}}{g} \sin 2 \theta\) (D) \(\frac{2 u^{2}}{g} \tan \theta\)

A glass wind screen whose inclination with the vertical can be changed is mounted on a car. The car moves horizontally with a speed of \(2 \mathrm{~m} / \mathrm{s}\). At what angle \(\alpha\) with the vertical should the wind screen be placed so that rain drops falling vertically downwards with velocity \(6 \mathrm{~m} / \mathrm{s}\) strike the wind screen perpendicularly? (A) \(\tan ^{-1}\left(\frac{1}{3}\right)\) (B) \(\tan ^{-1}(3)\) (C) \(\cos ^{-1}(3)\) (D) \(\sin ^{-1}\left(\frac{1}{3}\right)\)

On a two-lane road, car A is travelling with a speed of \(36 \mathrm{~km} \mathrm{~h}^{-1}\). Two cars \(\mathrm{B}\) and \(\mathrm{C}\) approach car \(\mathrm{A}\) in opposite directions with a speed of \(54 \mathrm{~km} \mathrm{~h}^{-1}\) each. At a certain instant, when the distance \(\mathrm{AB}\) is equal to \(\mathrm{AC}\), both being \(1 \mathrm{~km} . \mathrm{B}\) decides to overtake \(\mathrm{A}\) before C does. What minimum acceleration of car \(\mathrm{B}\) is required to avoid an accident? (A) \(3 \mathrm{~m} / \mathrm{s}^{2}\) (B) \(2 \mathrm{~m} / \mathrm{s}^{2}\) (C) \(1 \mathrm{~m} / \mathrm{s}^{2}\) (D) None
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