Having established rotational kinematics, it seems logical to extend our study of rotational motion to dynamics. Just as we began our study of Newtonian dynamics by defining a force, we start our study of rotational dynamics by defining our analogue to a force, the torque. From here, we will derive a general expression for the angular acceleration produced by a torque, which is quite similar to Newton's Second Law. We will also define a new concept, the moment of inertia of a rigid body.
Definition of a Torque
When we studied translational motion, a given force applied to a given particle always produced the same result. Because in rotational motion we consider rigid bodies rather than particles, we cannot make such a general statement about the effect of an applied force. For example, if the force is applied to the center of the object, it will not cause the object to rotate. If, however, it is applied to the edge of a rotating object, it can have quite a large effect on the rotation of the object. With this aspect of rotational motion in mind, we define torque to generally describe the effect a force will have on rotational motion.
Consider point P a distance r away from an axis of rotation, and a force F applied to P at an angle of θ to the radial direction, as shown below. If the force is parallel to the radius of the particle (θ = 0), then the force might cause some translational motion of the particle. But, since the force has no component acting in the tangential direction, it causes no change in rotational motion. In addition, if the force is close to the axis of rotation it will cause less change in the rotation of the body than at a farther distance. Thus we define the torque (denoted by τ) accordingly:
|τ||= Fr sinθ|
The second equation (τ = r×F) expresses the torque in terms of a cross product, an important operation in vector algebra, but not essential for the understanding of torque. With this vector definition, however, we are able to define the direction of the torque. The torque (because it is a cross product) must be perpendicular to both the force applied and the radius of the particle, implying that it points perpendicular to the plane of rotation of the particle.
This definition can be difficult to grasp conceptually, so we will consider some examples to clarify. The best example of a torque is the force applied to opening a door. The easiest way to open the door (in other words, the way to provide maximum torque) is to grab a point the furthest away from the hinges (like the handle), and pull perpendicular to the door itself. In this way, we give a maximum r, and sinθ = 1. The closer to the hinges one pulls, the more force must be exerted to provide the same torque on the door. In addition, the angle with which the torque is applied changes the force necessary for a given torque. The case of pulling perpendicular to the door requires the least force.
Torque plays the same role in rotational motion as force plays in translational motion. In fact, we can restate Newton's First Law to make it apply to rotational motion:
If the net torque acting on a rigid object is zero, it will rotate with a constant angular velocity.
Though this statement helps us to gain a conceptual understanding of exactly how a torque influences rotational motion, we need a rotational analogue to Newton's Second Law, which will serve as a quantitative basis for rotational dynamics.