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### Fundamentals Of Physics

Book edition 10th Edition
Author(s) David Halliday
Pages 1328 pages
ISBN 9781118230718

# A pulley wheel that is ${\text{8.0 \hspace{0.17em}cm}}$ in diameter has a ${\text{5.6 m}}$ long cord wrapped around its periphery. Starting from rest, the wheel is given a constant angular acceleration of ${\mathbf{1}}{\text{.}}{\mathbf{5}}{\mathbf{}}{\text{\hspace{0.17em}}}{\mathbf{rad}}{\mathbf{/}}{{\mathbf{s}}}^{2}{\mathbf{}}$. (a) Through what angle must the wheel turn for the cord to unwind completely? (b) How long will this take?

a) The angle that the wheel must turn for the cord to unwind completely is .$140\text{\hspace{0.17em}}\text{rad}$

b) Time taken by wheel to turn for the cord to unwind completelyis.$14\text{sec}$

See the step by step solution

## Step 1: Given

i) Diameter of pulley is $D=8.0\text{\hspace{0.17em}}\text{cm}$,

ii) Length of cord is $L=5.6\text{\hspace{0.17em}}\text{m}$,

iii) Angular acceleration is $\alpha =1.5\text{\hspace{0.17em}}\text{rad}/{s}^{2}$,

## Step 2: Determining the concept

Find the circumference of the wheel. Using circumference and length of cord, find the number of revolutions. Find the angle using the revolutions. Then, use rotational kinematic equations to find the time.

The circumference of the pulley,

${\mathbf{c}}{\mathbf{=}}{\mathbf{\pi d}}$

The number of revolutions made,

${\mathbf{\theta }}{\mathbf{=}}{\mathbf{2}}{\mathbf{\pi n}}$

Total angular distance travelled,

${\mathbf{\theta }}{\mathbf{=}}{\mathbf{2}}{\mathbf{\pi n}}$

Relation between angular distance and time taken,

${\mathbf{\theta }}{\mathbf{=}}{{\mathbf{\omega }}}_{0}{\mathbf{t}}{\mathbf{+}}\frac{1}{2}{{\mathbf{\alpha t}}}^{2}$

where, Iisthe moment of inertia, M, m are masses, r is radius, t is time taken.

## Step 3: (a) Determining the angle must the wheel turn for the cord to unwind completely

Circumference of wheel is c,

$c=\pi d$

$c=\pi ×0.08\text{\hspace{0.17em}m}$

$c=0.25\text{\hspace{0.17em}}\text{m}$

Now, number of revolutions n as follows:

$n=\frac{\text{Length of cord}}{\text{circumference of wheel}}$

$n=\frac{5.6\text{\hspace{0.17em}m}}{0.25\text{\hspace{0.17em}m}}$

$n=22.3\text{\hspace{0.17em}}\text{revolutions}$

Now, angle is as follows:

$\theta =2\pi n$

$\theta =2\pi ×22.3$

$\theta =140\text{\hspace{0.17em}}\text{rad}$

Hence,the angle that the wheel must turn for the cord to unwind completely is $140\text{\hspace{0.17em}}\text{rad}$.

## Step 4: (b) Determining how long wheel turn for the cord to unwind completely

Now, time can be calculated using the equation-

$\theta ={\omega }_{0}×t+\frac{1}{2}\alpha {t}^{2}$

$140\text{\hspace{0.17em}rad}=0×t+\frac{1}{2}×1.5\text{\hspace{0.17em}m}/{\text{s}}^{\text{2}}×{t}^{2}$

$t=13.7\text{sec}$

In two significant figures,

$t=14\text{s}$

Hence, time taken by wheel to turn for the cord to unwind completelyis $14\text{\hspace{0.17em}sec}$.