Before discussing the more complex case of rigid body rotation, it is instructive first to discuss the case of a particle of mass m rotating about some fixed point under the influence of external force F.

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A simple rigid body, free to rotate about an axis through O, consists of a particle of mass m  fastened to the end of a rod of length r  and negligible mass. An applied force F causes the body to rotate.

Tangential component of the force provides a tangential acceleration at , and Ft = m at . The torque about the center of the circle due to Ft is

t = Ft r = (m at) r = (m r a) r = m r2 a = I a .

That is, the torque acting on the particle is proportional to its angular acceleration, and the proportionality constant is the moment of inertia.  It is important to note that tI a is the rotational analogue of Newton's second law motion, F = ma.

Although we derived tI a for the special case of a single particle rotating about a fixed axis, it holds for any rigid body rotating about a fixed axis, because any such body can be analyzed as an assembly of single particles. Although each point on a rigid body rotating about a fixed axis may not experience the same force, linear acceleration, or linear velocity, each point experiences the same angular acceleration and angular velocity an any instant. Therefore, at any instant the rotating object as a whole is characterized by specific values for angular acceleration, net torque, and angular velocity.

If you see an object rotating, is there necessarily a net torque acting on it?

Sample problem
Rotational Motion EXP. 5:Rotational Motion

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