Answered: A particle moves along a circular path…

Principles of Physics: A Calculus-Based Text

5th Edition

ISBN:9781133104261

Author:Raymond A. Serway, John W. Jewett

Publisher:Raymond A. Serway, John W. Jewett

Chapter3: Motion In Two Dimensions

Section: Chapter Questions

Problem 31P: Figure P3.31 represents the total acceleration of a particle moving clockwise in a circle of radius...

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Question

A particle moves along a circular path of radius300 mm. If its angular velocity isθ = (3t2) rad/s, where t isin seconds, determine the magnitudes of the particle’svelocity and acceleration whenθ = 45°. The particle startsfrom rest when θ= 0°

Expert Solution

Step 1

Given

The radius of the circle is $r = 300 mm = 300 \times 10^{- 3} m = 0.3 m$ .
The angle $θ = 45 ° \times \frac{π}{180 °} = 0.785 rad$ .
The angular velocity is $\dot{θ = ω} = 3 t^{2} rad / s$ .

The angular acceleration is calculated as,

$\begin{array}{rcl} α & = & \frac{d \dot{θ}}{d t} \\ = & \frac{d (3 t^{2})}{d t} \\ = & 3 \times 2 t \\ = & 6 t \end{array}$

The angular position is calculated as,

$\begin{array}{rcl} θ & = & \int \dot{θ d t} \\ = & \int 3 t^{2} d t \\ = & 3 \times \frac{t^{3}}{3} \\ = & t^{3} \end{array}$

Substitute the known values.

$\begin{array}{rcl} 0.785 rad & = & t^{3} \\ t & = & \sqrt[3]{0.785} \\ = & 0.9225 s \end{array}$

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