Rotational Motion In this section, we will develop the relationship between torque and angular acceleration. You will need to have a basic understanding of moments of inertia for this section.
Imagine a force F acting on some object at a distance r from its axis of rotation. We can break up the force into tangential (Ftan), radial (Frad) (see Figure 1). (This is assuming a two-dimensional scenario. For three dimensions -- a more realistic, but also more complicated situation -- we have three components of force: the tangential component Ftan, the radial component Frad and the z-component Fz. All components of force are mutually perpendicular, or normal.)

From Newton's Second Law,

    Ftan = m atan
However, we know that angular acceleration, , and the tangential acceleration atan are related by: 
    atan = r 
Then,
    Ftan = m r 
If we multiply both sides by r (the moment arm), the equation becomes
    Ftan r = m r
Note that the radial component of the force goes through the axis of rotation, and so has no contribution to torque. The left hand side of the equation is torque. For a whole object, there may be many torques. So the sum of the torques is equal to the moment of inertia (of a particle mass, which is the assumption in this derivation), I = m r multiplied by the angular acceleration, .

Figure 1 Radial and Tangential Components of Force, two dimensions


Figure 2 Radial, Tangential and z-Components of Force, three dimensions

If we make an analogy between translational and rotational motion, then this relation between torque and angular acceleration is analogous to the Newton's Second Law. Namely, taking torque to be analogous to force, moment of inertia analogous to mass, and angular acceleration analogous to acceleration, then we have an equation very much like the Second Law.


Example Problem on Torque and Angular Acceleration
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