Answered: a) the angular acceleration b) the…

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

Starting from rest, a 4-wheel car accelerates for 20 s with a constant linear acceleration of 0.8 m/s2.
The radius of the tires is 0.4 m.
Find: a) the angular acceleration
b) the angle through which each wheel has rotated.

Expert Solution

Step 1

Angular Acceleration

The angular acceleration of a rotating body is defined as the rate of change of angular velocity.

$α = \frac{∆ ω}{∆ t}$

The SI unit of angular acceleration is radians per second or $r a d / s$ .

For no slipping condition, the angular momentum of a rigid body is related to its linear acceleration $a$ by the relation

$a = α R \Rightarrow α = \frac{a}{R}$

$R$ is the radius of the object.

If $θ$ is the angular displacement of a rotating object in time $t$ with an initial velocity $ω_{i}$ , then

$θ = ω_{i} t + \frac{1}{2} α t^{2}$

Step 2

a) In the given question, the linear acceleration of the tires $a = 0.8 m / s^{2}$

the time for which the wheel is rotated $t = 20 s$

the radius of the tires $R = 0.4 m$

The angular acceleration of the tires

$α = \frac{a}{R} = \frac{0.8}{0.4} r a d / s^{2} = 2 r a d / s^{2}$

b) The car has started from rest, then the initial angular velocity is equal to zero.

$ω_{i} = 0$

If $θ$ is the angular displacement of the tires then

$θ = ω_{i} t + \frac{1}{2} α t^{2}$

Thus the angular displacement of the tires in $t = 20 s$ is

$θ = \frac{1}{2} \times 2 \times {(20)}^{2} = 400 r a d$

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