Question

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

Short answer

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

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

Step-by-step solution

Step 1: Given

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

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

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

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.

(a) Determining the angle must the wheel turn for the cord to unwind completely

Circumference of wheel is c,

$c=\pi d$

$c=\pi \times 0.08\text{\hspace{0.17em}m}$

$c=0.25\text{m}$

Now, number of revolutions n as follows:

$n=\frac{\text{Lengthofcord}}{\text{circumferenceofwheel}}$

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

$n=22.3\text{revolutions}$

Now, angle is as follows:

$\theta =2\pi n$

$\theta =2\pi \times 22.3$

$\theta =140\text{rad}$

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

(b) Determining how long wheel turn for the cord to unwind completely

Now, time can be calculated using the equation-

$\theta ={\omega }_0\times t+\frac{1}{2}\alpha t^2$

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

$t=13.7\text{sec}$

In two significant figures,

$t=14\text{s}$

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