NEWTON'S SECOND LAW FOR
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.
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
t = I a
is the rotational analogue of Newton's second law motion, F = ma.
we derived t = I
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?