Help Basic Rotational Dynamics Question

[4getr34]

Can anyone help me figure out this question? I swear this isn't homework! I am actually from a car forum and would like to understand some basic principles behind car engine pulleys.

Pulley A drives another pulley Bx via a belt.

Pulley A
diameter:15inches
weight: 1lbs
applied torque: 10lbs*1inch

Pulley B1
diameter: 5inches
weight: 0.5lbs

Pulley B2
diameter: 5inches
weight: .75lbs

Pulley B3
diameter: 7inches
weight: .25lbs

Pulley B4
diameter: 7inches
weight: .75lbs

Which of the Bx pulleys will require more torque from Pulley A to rotate? How do you figure this out? What is the resulting torque on each of the Bx pulleys?

thanks a lot!

 

[K^2]

Assuming all pulleys are geometrically similar, here is how you go about it.

Belts ensure that surface speed of all pulleys is identical. Id est,

$$R_A \omega_A = R_i \omega_i$$

And consequently,

$$R_A \dot{\omega}_A = R_i \dot{\omega}_i$$

And of course, the equation for angular acceleration and torque,

$$I_i \dot{\omega}_i = \tau_i$$

The only problem is that there is no specific equation for moment of inertia, I_i for pulleys. It's going to be close to (1/2)MR² for solid cylinder, but it can be a little higher or lower depending on the geometry. This is where argument for similar geometries should come in. If all pulleys have similar geometries, then

$$I_i = c M_i R_i^2$$

What that c is, doesn't really matter. It's a dimensionless constant. The important bit is that it should be the same for all pulleys. Using that, you should be able to express torque for each pulley in terms of angular acceleration of A, masses and radii of the pulleys, and this coefficient c. Then compare results. Good luck.

 

[4getr34]

wow thanks a lot for the reply!

so if i understand correctly as long as the geometries on the pulleys are similar its basically I = MR^2

B1
I = 3.125 lbs * inch^2
B2
I = 4.6875 lbs * inch^2
B3
I = 3.0625 lbs * inch^2
B4
I = 9.1875 lbs * inch^2

i'm going to see if i can find out from the companies actual geometries but this is really interesting as B4 is my impression of a overweight underdrive pulley and I'm actually trying to decide between B3(underweight oversize) and B1 (stock) pulleys. thank you very much!

 

[K^2]

If you actually need an estimate on torque, it's closer to (1/2)MR². Whether it's a little more or a little less depends on exact geometry.

 

[Abiruth]

Check this site for a clearer picture

http://library.thinkquest.org/27948/pulley.html

hope it helps