where ri is the perpendicular distance between a particle of mass mi and the rotation axis. The rotational inertia depends on the location of the rotation axis.
Torque measures the effectiveness of a force for twisting or turning an object. It can be calculated in two equivalent ways: either as the product of the perpendicular component of the force with the shortest distance between the rotation axis and the point of application of the force
or as the product of the magnitude of the force with its lever arm (the perpendicular distance between the line of action of the force and the axis of rotation)
A force whose perpendicular component tends to cause rotation in the CCW direction gives rise to a positive torque; a force whose perpendicular component tends to cause rotation in the CW direction gives rise to a negative torque.
The work done by a constant torque is the product of the torque and the angular displacement:
W = τθ (θ in radians)
(8-6)
The conditions for equilibrium are
Fnet = 0 and τnet = 0
(8-8)
The rotation axis can be chosen arbitrarily when calculating torques in equilibrium problems. Generally, the best place to choose the axis is at the point of application of an unknown force so that the unknown force does not appear in the torque equation.
Newton's second law for rotation is
Στ = Iα
(8-9)
where radian measure must be used for α. A more general form is
The total kinetic energy of a body that is rolling without slipping is the sum of the rotational kinetic energy about an axis through the center of mass and the translational kinetic energy:
The angular momentum of a rigid body rotating about a fixed axis is the rotational inertia times the angular velocity:
L = Iω
(8-14)
The law of conservation of angular momentum: if the net external torque acting on a system is zero, then the angular momentum of the system cannot change.
If τnet = 0, Li = Lf
(8-15)
The following table summarizes the analogous quantities in translational and rotational motion.
