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Chapter 10: Q15P (page 288)

Starting from rest, a wheel has constant acceleration $伪 = 3.0 \frac{rad}{s^{2}}$ . During a certain role="math" localid="1660898447415" $4.0 s$ interval, it turns through $120 rad$ . How much time did it take to reach that $4.0 s$ interval?

Short Answer

Expert verified

Wheel requires $8.0 s$ time to reach that interval.

Step by step solution

01

Listing the given quantities

The angular acceleration of a wheel, $伪 = 3.0 鈥� \frac{rad}{s^{2}}$

The time interval of $120 rad$ displacement is, $t = 4.0 s$

The angular displacement is, $胃 = 120 鈥� rad$

02

Understanding the kinematic equations

Use the kinematic equation for constant angular acceleration to calculate the initial angular velocity of the given interval. Use angular velocity formula to determinethefinal angular velocity asthewheel starts from rest.

Formula:

i) $蝇 = 蝇_{0} + 伪迟$

ii) $胃 = 蝇_{0} t + \frac{1}{2} {伪迟}^{2}$

03

Calculate the angular velocity

The kinematic equation for angular motion is:

$胃 = 蝇_{0} t + \frac{1}{2} 伪 t^{2}$

Substitute the value and solve as:

Solving for $蝇_{0}$ :

$蝇_{0} = 24 鈥� \frac{rad}{s}$

04

Calculate the time required for the wheel to get the angular speed of 24 rads

Consider the expression for the angular speed as:

$蝇 = 蝇_{0} + 伪迟$

Solving for time, $t = T$

$T = \frac{蝇鈭� 蝇_{0}}{伪}$

Substitute the values and solve as:

$\begin{matrix} T = \frac{24 鈭� 0}{3.0} \\ = 8.0 s \end{matrix}$

Time required for the wheel is $8.0 s$ .

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