MHB Linear and angular velocity of 2 pulleys and a belt.

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A belt connects two pulleys with radii of 5 inches and 3 inches, with the larger pulley rotating at 1000 revolutions per minute. The linear velocity of the belt is calculated to be approximately 43.63 feet per second. The smaller pulley, with a radius of 3 inches, is determined to be rotating at about 1667 revolutions per minute. The calculations confirm that the methods used for determining both the linear speed of the belt and the angular velocity of the smaller pulley are accurate. This discussion highlights the relationship between the radii of the pulleys and their respective rotational speeds.
karush
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A belt contects two pulleys with radii $\displaystyle5\text { in}$ and $3\text { in}$

the $\displaystyle5\text { in}$ pulley is rotating at $\displaystyle\frac{1000\text{ rev}}{\text{min}}$

What is the linear $\displaystyle\text{v}$ in $\displaystyle\frac{\text{ft}}{\text{sec}}$ of the belt?

$\displaystyle \text{v}=
\frac{1000\text{rev}}{\text{min}}
\cdot\frac{\text{min}}{60\text{ sec}}
\cdot\frac{10\pi \text{ in}}{\text{rev}}
\cdot\frac{\text{ ft}}{12 \text{in}}
=\frac{125\pi\text{ ft}}{9\text{sec}}
=43.63\frac{\text{ft}}{\text{sec}}
$

How many revolutions per min is the $\text{3 in}$ pulley making?

so
$\displaystyle \omega_{3in}
=\frac{5}{3}
\cdot\frac{1000\text{rev}}{\text{min}}
\approx 1667\frac{\text{rev}}{\text{min}}$

no ans in bk on this so hope ans here is perhaps it.
 
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To find the linear speed of the belt, we may state (using the information about the larger pulley):

$$v=r\omega=\left(5\text{ in}\frac{1\text{ ft}}{12\text{ in}} \right)\left(1000\frac{\text{rev}}{\text{min}} \frac{2\pi\text{ rad}}{1\text{ rev}} \frac{1\text{ min}}{60\text{ s}} \right)=\frac{125}{9}\pi\frac{\text{ ft}}{\text{s}}$$

This agrees with your result, although I think the way I have written the intermediary steps makes it a bit clearer what is going on. (Bandit)

Now, to find the revolutions per minute of the smaller pulley (the second pulley), we may write (as we did in your previous topic):

$$\omega_2=\frac{r_1}{r_2}\omega_1$$

You have done this correctly as well. (Clapping)
 
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