MadSci Network: Engineering

Re: Static vs. Dynamic Torque

Date: Fri Sep 4 04:01:30 1998
Posted By: Justin Roux, Other (pls. specify below), Grad, professional engineer plus research, Intergraph UK Ltd
Area of science: Engineering
ID: 902193942.Eg

Hello Irene.

How cruel you are. A question on the mathematical principles behind 
mechanics, and that sends even the hardest scientists running in fright. I 
had to teach myself this again just to answer your question. Here we go.

Torque is the product of Force and radius. T=Fr and that covers static 
torque in just a moment (excuse the pun)
Now, to dynamics.
Force = mass x acceleration and acceleration is rate of change of velocity 
dv/dt so
F = m dv/dt
In rotational velocity, the speed of a point on the rotating body is a 
product of w (frequency) and the radial distance from the centre. If we 
apply that to the above equation we get
F = m r dw/dt
Now put that back into our torque equation T=Fr to give
T = mr^2 dw/dt which is a general case.

In physics we call the integrated summation of mr^2 the rotational moment 
of inertia ( I )and quote the above formula empirically as :
Dynamic Torque T = Fr = d (Iw)/dt which I think is the correlation you 
were looking for. I notice in your question that you mentioned energy. Do 
not confuse Energy with Torque. Use the kinetic energy equation E = mv^2/2 
in a similar manner to the above get the rotational kinetic energy 
equation E = Iw^2/2.

Now, since it is only fitting that you delve further into beautiful 
mathematics (as a punishment!), I have added some alternative values for 

I = mr^2/2 for a disc about a perpendicular axis through the centre.
I = mr^2/4 for a disc spinning about its diameter
I = 2mr^2/3 for a hollow sphere about its diameter
I = 2mr^2/5 for a solid sphere about its diameter.


Justin Roux

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