|MadSci Network: Engineering|
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'. 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. Enjoy. Justin Roux
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