Help Basic Rotational Dynamics Question

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The discussion focuses on understanding the torque requirements for various pulleys driven by a primary pulley in a car engine setup. The key principle is that the surface speed of all pulleys must remain constant, leading to equations that relate angular acceleration and torque. Moment of inertia for each pulley can be approximated using the formula I = (1/2)MR², with adjustments for specific geometries. Calculated moments of inertia for the Bx pulleys reveal varying torque needs, with B4 showing the highest inertia, indicating it requires more torque to rotate. The conversation emphasizes the importance of pulley geometry in determining performance characteristics.
4getr34
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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!
 
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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, Ii 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.
 
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!
 
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.
 

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