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Rotational motion. Radians, and kinematics of rotational motion.	Rotational dynamics: Torque, moment of inertia, rotational kinetic energy, conservation of angular momentum.	Statics: Forces and torques on objects in equilibrium

In the last section, we learned about the new method of angular measurement, radians, and the language for describing rotational motion. In the remainder of this part of the course, we develop dynamics, energy, a new momentum quantity, angular momentum, and finally conservation of angular momentum. These topics required seven chapters for the corresponding linear or translational quantities. Don't be surprised if things go too quickly, but remember that all these arguments follow from the physics you have already seen. There is actually very little new here:

Uniform angular motion and uniform angular accelerated motion

For example, we have already seen that motion in the presence of a uniform acceleration in one direction is described by the equations:

These equations can immediately be used to tell us about angular position and angular velocity in uniformly tangentially accelerated circular motion. Recall that the angular acceleration, speed around the circle, and angular position are all related to the regular translational quantities by multiplying by the radius of the circle:

Therefore, after substituting these results into the equations for uniform accelerated motion, we find (after canceling all the factors of radius) that motion with uniform angular acceleration is given by:

We will use this approach of taking known equation and substituting in the position, speed, and acceleration (in terms of angular quantities and radius of circle) over and over to derive new equations for the angular quantities. After a bit of practice, you can see that only ONE set of result needs to be in your head. The remainder can be derived.

Torque and angular acceleration: Dynamics

At this point, we have the kinematics of angular motion under control. Now we are ready to move on to the process of PREDICTING angular accelerations and resulting motion. Our principal goal is to determine what it is that leads to angular acceleration.

Experience shows us that forces applied to an object might, or might not lead to changes in the rotational behavior of the object. Forces that point directly at or directly away from the axis of rotation do not lead to changes in the rotation. However, forces that point so that the direction of the force misses the axis CAN AND DO change the rotation. Such forces are said to have a lever arm. Lever arm is defined by drawing a line in the direction of the force, and measuring the distance of closest approach this line makes to the axis:

We refer to the force multiplied by the perpendicular distance to the axis, torque (rhymes with dork).

Torque is the quantity that changes rotational motion. For rotations, it is the object that replaces force as the cause of angular acceleration.

Newton's Laws for rotations

As we have seen, torque is the quantity that changes rotational motion. Now we need to derive a precise rule for calculating exactly HOW the torque can be used to predict angular acceleration. The first thing we state is a version of Newton's 1^st Law:

Things rotating remain in their state of uniform rotational velocity unless acted on by net torque.

Similarly, once we observe angular acceleration, we know that net torque is acting. We need a rule to relate net torque to angular acceleration. We start with Newton's 2^nd Law:

Now, imagine a point mass rotating around a circle and think about the tangential acceleration:

Now multiply each side by the radius of the circle:

The left side is just the torque. The right side has the tangential acceleration, which is just the angular acceleration multiplied by the radius of the circle:

We define a new quantity that depends upon the mass and location of the mass from the axis of rotation. We call it the moment of inertia.

Therefore, net torque leads to angular acceleration, but we need the moment of inertia to tell us how large an acceleration results. Moment of inertia in rotational problems takes the place that mass has in translational problems.

Now put all the pieces together:

Rotational kinetic energy

Now, we move on to develop the ideas of energy for rotational motion. We have already seen that particles have kinetic energy due to motion. However, the speed of a given particle is due to combination of the center of mass motion and the rotations. Therefore, the 1/2mv^2 has two pieces. Therefore, the kinetic energy of an object that is both translating and rotating about an axis is :

What IS the kinetic energy of rotation? Again, think about two point masses on opposite sides of a circle of radius, r, and rotating about the circle:

The center of mass of this object is at the center of the circle and is NOT translating. Therefore, the KE of this system is entirely rotational. Each mass has the same speed. Therefore, the total KE is:

Therefore, the total KE for an object is:

When doing energy problems, conversion of potential energy to KE must include the correct kinetic INCLUDING rotational kinetic.

Example: Rolling without slipping

Conservation of energy is as effective in rotational problems as it is in linear motion. Many times, we can avoid nasty torque calculations by using energy conservation. As an example of the use of rotational kinetic energy, let's discuss the process of rolling without slipping. Here is a picture of a rolling object:

A circular object (hoop, disk, sphere, etc) can roll down an inclined plane. If the point of contact does not slip, then the motion of the center of the circle and the motion of the circumference are related by:

Therefore, the total KE is given by:

Therefore, depending upon the moment of inertia, we find that different circular objects reach the bottom of the hill with different center of mass velocities.

Conservation of angular momentum

Just as for the linear case, we can rewrite the torque equation in terms of a change in time of a type of momentum, angular momentum:

where the angular momentum is defined by:

Then, in the absence of external torque, L is conserved, by just the same arguments that we used in the case of linear momentum. Conservation of angular .

Important points

• Object will remain in a state of uniform rotation at fixed angular velocity unless acted on by a net force.
• Net torque leads to angular acceleration via , where the moment of inertia is given by adding up all the mass, multiplied by the square of the radius each piece has from the axis of rotation: .
• Rotation gives each mass of the object a velocity due to the rotation. Therefore, rotations lead to kinetic energy of the form,
• We can also define an angular momentum,

such that net torque causes a change in angular momentum. In the absence of net torques from the outside, angular momentum is conserved.