Circular and Rotational Motion Torque and the Moment of Inertia

The simplest way to consider the motion of some point mass along a circular trajectory is begin with Newton’s first law: that an object in motion remains in that motion, in the same direction, unless acted upon by an outside force.  Since the motion of an object along a circular trajectory implies a motion in a continuously changing direction (coming back to point the same way only once per revolution), this law demands a continually applied force.  This force, the centripetal force, is given by

F = m v² / R,

where m is the mass of the object, v its velocity and R the radius of it’s circular trajectory.

With this basic analysis, you can already answer a physics question: suppose someone is spinning a ball of mass 0.1 kg at the end of a string along a circular trajectory such that the ball is moving with a speed of 10 m/s and such that the distance from the ball to the person’s hand along the string is one meter.  Using the equation for the centripetal force, we can then find the force that the string must exert on the ball to keep it in circular motion (the string tension): 10 N.

Rather than shoehorning circular motion into the machinery of straight-line mechanics, as was done above, it is often easier to employ a series of rotational quantities analogous to the usual straight line quantities.  The new quantities that will be used:

Angular position (θ): Considering the origin as the center of the circular motion, any point along the trajectory can be described simply by an angle from some reference point on the trajectory.  Thus, as the object moves along the trajectory, it sweeps out angles from 0 degrees to 360 degrees repeatedly.  These angles are often written in radians rather than degrees (0 to 2*Pi instead of 0 to 360).

Angular velocity (ω): Akin to the angular position, the angular velocity is written as some amount of angle through which the object passes in some time (rad/s), the time-derivative of angular position.

Angular acceleration (α): The rate of change of angular velocity (second time-derivative of angular position).

Moment of inertia (I): A quantity that describes both the mass of an object and the radial distribution of that mass and is given by

I = m R²,

where m is the mass and R is the distance from the center of rotation.  Thus a 1 kg mass will have a moment of inertia of 1 kg m² if it is 1 m away from the center of rotation, but 4 kg m² if it is 2 m away.

Torque (N): A quantity that describes both the force applied to a rotating body and the position at which the force is applied and that is given by the vector equation

N = r x F,

where r is the vector pointing radially from the center of rotation to the point where the force is applied and F is the force vector.  In the simple case where r and F are perpendicular (where the force is applied in the direction of circular motion), this reduces to just

N = r F.

Having defined these quantities, it is then possible to write a purely rotation analog to Newton’s second law, in which the force is replaced by torque, the mass by the moment of inertia and the acceleration by the angular acceleration, taking the form:

N = I α.

One can also write down an expression for the rotational kinetic energy (K):

K = I ω²

In this more sophisticated treatment, one is not limited to considering the motion only of point masses at a given radius; the moment of inertia can be computed for any object about the center of rotation, e.g. disc, sphere, rod, etc.  This can be done either by referencing a table of common moments of inertia or by computing the integral of the object’s mass density times the radius squared over the extent of the object.