Answered: An object is moving around a circle of…

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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Question

Let θ =23π/4

b.) What angle in the interval [0, 2π) is coterminal with θ? Express this angle in radians.

c. ) An object is moving around a circle of radius 8 such that it rotates by θ in 2 minutes.

i.) Determine how far the object moves along the circumference of the circle.

ii.) Determine the object’s angular and linear speed.
.

Expert Solution

Step 1 :Introduction

The linear displacement is calculated by using the formula $d = r θ$ , where $r = 8 m$ is the radius of the circle and $\begin{array}{rcl} θ & = & \frac{23 π}{4} \\ = & (1035 \times \frac{π}{180}) r a d \\ \approx & 18.1 r a d \end{array}$ is the angular displacement.

The angular velocity is calculated by using the formula $ω = \frac{θ}{t}$ , where $\begin{array}{rcl} θ & \approx & 18.1 r a d \end{array}$ is the angular displacement and $\begin{array}{rcl} t & = & 2 m i n \\ = & 2 \times 60 s \\ = & 120 s \end{array}$ is the time taken .The linear velocity is calculated by using the formula $v = \frac{d}{t}$ , where d is the linear displacement and $\begin{array}{rcl} t & = & 120 s \end{array}$ is the time taken .

Step 2 :Calculation of the linear displacement

Calculate the linear displacement by using the formula $d = r θ$ , where $r = 8 m$ is the radius of the circle and $\begin{array}{rcl} θ & \approx & 18.1 r a d \end{array}$ is the angular displacement.

$\begin{array}{rcl} d & = & r θ \\ = & (8 m) \times (18.1 r a d) \\ = & 144.8 m \end{array}$

Step 3 :Calculation of the angular velocity

Calculate the angular velocity by using the formula $ω = \frac{θ}{t}$ , where $\begin{array}{rcl} θ & \approx & 18.1 r a d \end{array}$ is the angular displacement and $\begin{array}{rcl} t & = & 120 s \end{array}$ is the time taken .

$\begin{array}{rcl} ω & = & \frac{θ}{t} \\ = & \frac{18.1 r a d}{120 s} \\ \approx & 0.15 r a d / s \end{array}$

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