Angular Velocity: Pulley and belt system

[thatstheguy9]

Homework Statement

Determine the angular velocity of Pulley B and C

Relevant Equations

$V = \omega r$

So far I have:

The velocity of the belt will be the same for pully A and D, so we can calculate the angular velocity of pulley D:

$V_A = V_B$
$\omega_A r_A = \omega_D r_D$
$((20*3)+40)(0.075) = \omega_D (0.025)$
$\omega_D = 300 Rad/s$

My next step was to determine the angular velocity of pulley B. My thought was because pully B and pulley D are joined on the same shaft they must have the same velocity.
I determined $\omega_B = 75 Rad/s$.
However the solution found $\omega_B = 300 Rad/s$.
I don't understand how the larger pulley can have a higher velocity than the smaller pulley. Or why the velocity of pulley A is the same as pulley B. Can someone explain this?

 

[haruspex]

Homework Statement:: Determine the angular velocity of Pulley B and C
Relevant Equations:: $V = \omega r$

View attachment 290561

So far I have:

The velocity of the belt will be the same for pully A and D, so we can calculate the angular velocity of pulley D:

$V_A = V_B$
$\omega_A r_A = \omega_D r_D$
$((20*3)+40)(0.075) = \omega_D (0.025)$
$\omega_D = 300 Rad/s$

My next step was to determine the angular velocity of pulley B. My thought was because they are joined on the same shaft they must have the same velocity.
I determined $\omega_D = 75 Rad/s$.
However the solution found $\omega_D = 300 Rad/s$.
I don't understand how the larger pulley can have a higher velocity than the smaller pulley. Or why the velocity of pulley A is the same as pulley B. Can someone explain this?
View attachment 290562

You seem to have got yourself confused.
You found $\omega_D = 300 Rad/s$, but then changed it to $\omega_D = 75 Rad/s$

 

[thatstheguy9]

You seem to have got yourself confused.
You found $\omega_D = 300 Rad/s$, but then changed it to $\omega_D = 75 Rad/s$

Apologies, I've corrected the original post.

 

[haruspex]

because they are joined on the same shaft

Which two are?

 

[thatstheguy9]

Which two are?

Pulley B and D

 

[haruspex]

Pulley B and D

Ok, so if D's angular velocity is 300r/s what is B's?

 

[thatstheguy9]

Ok, so if D's angular velocity is 300r/s what is B's?

I got 75 rad/s using:

$V_D = V_B$
$\omega_D r_D = \omega_B r_B$
$300*0.025 = \omega_B * 0.100$
$\omega_B = 75 rad/s$

 

[haruspex]

I got 75 rad/s using:
$V_D = V_B$

Ok, I see. You posted

My next step was to determine the angular velocity of pulley B. My thought was because pully B and pulley D are joined on the same shaft they must have the same velocity.

So I thought you meant they must have the same angular velocity. Which is true.

 

[thatstheguy9]

Ok, I see. You posted

So I thought you meant they must have the same angular velocity. Which is true.

How can they have the same angular velocity when they have different radi?
Which brings me to my next point of confusion:
Why they have used the radius of pulley D to calculate the angular velocity of pulley B.

 

[haruspex]

How can they have the same angular velocity when they have different radi?

They're attached rigidly to the same axle. If the axle completes one turn, how many turns, or what fraction of a turn, has each pulley turned?

 

[Lnewqban]

How can they have the same angular velocity when they have different radi?
...

Perhaps you are confusing tangential and angular velocitires.

 

[thatstheguy9]

I see where I went wrong. Thanks for your help @haruspex and @Lnewqban