Torque

At first we studied how to describe the linear motion of objects with the quantities: position, velocity, and acceleration. Then we studied the cause of linear motion as described by Newton's laws of motion. At this point we have even looked at how to describe the motion of objects that are rotating by using the quantities: angular position, angular velocity, and angular acceleration. Now we will begin to develop a model for why objects rotate or not.

It should be no surprise to you that a force must be applied to an object in order to get the object to start rotating. However, the location of the applied force, and the angle of the applied force also play a role in how fast the object begins to rotate. You are probably an expert at this concept already, what is the easiest way to open a door? Do you push further or closer away from the hinge, perpendicular to the door or parallel to the door? The mathematical way we deal with a force and its location on an object is via the quantity called torque. Torque is a vector and is found by taking the "cross product" (a mathematical operation) of the position from a reference axis that the force is applied and the force itself. This is mathematically written as...

$\vec{\tau} = \vec{r} \times \vec{F}$

The magnitude of torque is...

$\left| \vec{\tau} \right| = \left| \vec{r} \right| \, \left| \vec{F} \right| sin(\theta)$

...where $\theta$ is the smallest angle between the position vector and the force vector when placed tail-to-tail. It is highly recommended that you construct a vector-operation diagram to determine the angle $\theta$.

The cross product is a mathematical operation that asks, "how perpendicular is one vector to the other?". Thus the magnitude of torque is often found in the following way...

Or a physical representation of the above two definitions looks like...

The vector nature of torque is beyond the scope of this class at the moment. However, we still need to define a rotational direction. The convention that we used in rotational kinematics is also used here. Counter-clock-wise (ccw) is positive and clock-wise (cw) is negative. The sign convention is added in by hand based on which direction the force would want the object to rotate around a chosen reference point.

Statics and dynamics

Now that we have defined a way to mathematically work with torque, let's revisit Newton's laws of motion as applied to the point particle model (linear motion) and also introduce its form in the rigid body model (rotational motion).

Statics and dynamics are terms used to describe the different scenarios that fall out of the two Newton's 2nd law equations above. The scenarios can be summarized by the table below. In this section we will focus mostly on the rotational portion since this is the newer topic being introduced.

A new quantity shows up when dealing with rotational dynamics, moment of inertia $I_o$, where the subscribe "o" refers you to the axis of rotation. The moment of inertia is analogous to mass in the point particle model. The mass in the point particle model plays the role of inertia, which we can view in the following way: the larger the mass (i.e. the larger the inertia) the harder it is to increase the translation motion of an object. Think about a massive object on a frictionless surface, if you stand on a surface with friction and push the massive object, it is really hard to start to get moving at a high speed. If the object were much less massive, you would have no problem getting the object up to a fast speed. Now we can understand what role the moment of inertia plays in rotational motion. Just like mass, the larger the moment of inertia, the harder it is to increase the rotational motion of an object about an axis. Again, a simplified thought experiment goes something like this: if object that is not rotating has a large moment of inertia, it would require a lot of effort on your part to get it to start rotating at a fast angular speed. If the object has a much smaller moment of inertia, it would be much easier to get it rotating at a fast speed.

The moment of inertia depends on the axis of rotation and the way that the mass is distributed around this axis of rotation. For example, if you hold weights in our hands and stretch your arms out you would have a larger moment of inertia about the axis that runs from your feet to your head than you would if you held the weights tight to your body.

Now, take a look at the pre-lecture reading and videos below.