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An interaction which produces rotation.  

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h2. Motivation for Concept

Forces applied to a body in an attempt to produce rotation will have different effects depending upon both their direction of application and their location of application.  Most people have either accidentally or purposely experimented with opening a door by applying a force near the hinges and found it to be an inefficient procedure.  Obtaining rotation is much easier when the force is applied far from the hinges (hence the placement of door handles opposite the hinges).  

Location alone, however, is not enough to guarantee effective rotation.  Consider another experiment.  Suppose that you open a door so that it is ajar.  Position yourself at the edge of the door opposite the hinges and push directly along the door toward the hinges.  The door will not rotate, even with a hard push.  This indicates that the direction of the force is also important to the rotation produced.

With these experiments in mind, we recognize that we must define a new quantity that describes the effectiveness of an interaction at producing rotation about a specific axis (in our examples, the axis was set by the line of the door hinges).  This quantity is called torque.

h2. Conditions for One-Dimensional Torque

In introductory physics, it is sufficient (with the exception of certain special cases like the [gyroscope]) to consider torques in one dimension.  To ensure that the torques are one-dimensional, a situation must obey certain restrictions.

h4.  Motion in a Plane

To ensure that one-dimensional torques result, all objects in the system should have their center of mass confined to move in a plane, called the _xy_ plane.  Further, all forces should be applied such that their vectors lie in the _xy_ plane. 

h4.  Axis of Rotation Along z-Axis

Further, the axis of rotation chosen for the system must be perpendicular to the _xy_ plane, and so parallel to the z-axis. 

h2. Definition of Torque

h4. Cross Product

Defining the torque resulting from a force requires two pieces of information:

# The force applied (magnitude and direction).
# The position (magnitude and direction) of the force's application with respect to the [axis of rotation] about which the torque is to be calculated.

The torque is most succinctly defined by using the vector [cross product]:

{latex}\begin{large}\[\tau_{z} \equiv \vec{r}\times \vec{F} \]\end{large}{latex}

where τ is the torque, _r_ is the position of the point of application of the force with respect to the axis of rotation, and _F_ is the force.  Note that since the torque is assumed to lie in the +z or -z direction, we have specified its vector nature with the _z_ subscript rather than a vector arrow.

h4. Magnitude

In two dimensions, this formula is equivalent to:

{latex}\begin{large}\[ |\tau_{z}| = rF\sin\theta\]\end{large}{latex}

where the angle θ is the angle between the position vector and the force vector.  Some examples showing how to find θ are shown here.

PICTURE

h4. Direction

Note that this second form of the equation for torque is in terms of magnitude only.  To define the direction, the vectors _r_ and _F_ must be examined to determine the _sense_ of the rotation.  Since we are restricted to one-dimensional torques, we can describe the sense of the rotation as clockwise or counterclockwise.  Examples of forces producing clockwise and counterclockwise torques are shown here.

PICTURE

 When constructing free-body diagrams for systems in which torques are of interest, it is important to draw them from the perspective of someone looking along the axis of rotation and to assign a mathematical sign (+ or -) to each sense of rotation.  This is usually done as shown below.

PICTURE


h2. Parsing the Torque Magnitude

It is sometimes useful to associate the sinθ portion of the formula for the magnitude of the one-dimensional torque with either the force or the position.  These two possible associations lead to two terms that are often used in describing the rotational effects of a force: _perpendicular force_ and [lever arm].

h4. Perpendicular Force

Suppose we group the formula for the magnitude of the torque in the following way:

{latex}\begin{large}\[ |\tau_{z}| = r (F\sin\theta) \equiv r F_{\perp} \]\end{large}{latex}

where we have defined the perpendicular component of F as _F_ sinθ.  This name is chosen because, as shown in the pictures below, _F_sinθ is the size of the part of _F_ that is directed perpendicular to _r_.  

PICTURE

h4. Moment Arm

If we instead group the formula as:

{latex}\begin{large} \[ |\tau_{z}|= F (r \sin\theta) \equiv F r_{\perp} \] \end{large}{latex}

where we have defined a perpendicular component of r.  In this case, we give the component the special name of [lever arm].  The [lever arm] can be thought of as the closest distance of approach of the [line of action] of the force to the axis of rotation, as is shown in the pictures below.

PICTURES

The [lever arm] is a useful concept in solving physics problems, since it is common to be given the information to calculate the lever arm for a force without being told the force's exact position of application.  Some examples which require use of the lever arm are: