11: Angular Momentum and Static Equilibrium

*A helicopter has its main lift blades rotating to keep the aircraft airborne. Due to conservation of angular momentum, the body of the helicopter would want to rotate in the opposite sense to the blades, if it were not for the small rotor on the tail of the aircraft, which provides thrust to stabilize it.*

Angular momentum is the rotational counterpart of linear momentum. Any massive object that rotates about an axis carries angular momentum, including rotating flywheels, planets, stars, hurricanes, tornadoes, whirlpools, and so on. The helicopter shown in the picture can be used to illustrate the concept of angular momentum. The lift blades spin about a vertical axis through the main body and carry angular momentum. The body of the helicopter tends to rotate in the opposite sense in order to conserve angular momentum. The small rotors at the tail of the aircraft provide a counter thrust against the body to prevent this from happening, and the helicopter stabilizes itself. The concept of conservation of angular momentum is discussed later in this module. In the main part of this module, we explore the intricacies of angular momentum of rigid bodies such as a top, and also of point particles and systems of particles. But to be complete, we start with a discussion of rolling motion, which builds upon the concepts of the previous module.

In this module, we will continue our discussion about rotational motion and extend it to rotating about a moving axis. The first example of this type of motion is rolling. In the animation below, you can see that rolling is a combination of two types of motion: rotation + translation. We can use this idea to determine the kinetic energy of rolling, because it is literally K_{rolling} = K_{rotation} + K_{translation}.

We will also look into the forces that cause an object to roll without slipping. This motion occurs because there is a static frictional force at the contact point between the rolling object and the surface on which it rolls. We will draw a free-body diagram and apply Newton’s 2nd Law for translation and rotation to analyze the motion.

After this we will explore the concept of angular momentum. Remember how we related net force to changes in linear momentum? Well, there is an angular equivalent to this, where a net torque () causes a change in angular momentum (): Also, remember how we discussed if the net force acting on a system was zero, then the linear momentum of the system was conserved? Well, that’s also true for angular momentum. If the net torque acting on a system is zero, the angular momentum of the system is conserved.

In earlier modules, you learned about forces and Newton’s laws for translational motion. You then studied torques and the rotational motion of a body about a fixed axis of rotation. You also learned that static equilibrium means no motion at all and that dynamic equilibrium means motion without acceleration.

In this module, we combine the conditions for static translational equilibrium and static rotational equilibrium to describe situations typical for any kind of construction. What type of cable will support a suspension bridge? What type of foundation will support an office building? Will this prosthetic arm function correctly? These are examples of questions that contemporary engineers must be able to answer.

Finally, at the end of this module we will look at situations where the net force and net torque are both zero. This is our definition of mechanical equilibrium. When a system is in mechanical equilibrium and does not move, this is called static equilibrium. We will explore these static situations and determine why something doesn’t move, because there is always a reason why.

#### 11.1 Rolling Motion

- Describe the physics of rolling motion without slipping
- Explain how linear variables are related to angular variables for the case of rolling motion without slipping
- Find the linear and angular accelerations in rolling motion with and without slipping
- Calculate the static friction force associated with rolling motion without slipping
- Use energy conservation to analyze rolling motion

#### 11.2 Angular Momentum

- Describe the vector nature of angular momentum
- Find the total angular momentum and torque about a designated origin of a system of particles
- Calculate the angular momentum of a rigid body rotating about a fixed axis
- Calculate the torque on a rigid body rotating about a fixed axis
- Use conservation of angular momentum in the analysis of objects that change their rotation rate

#### 11.3 Conservation of Angular Momentum

- Apply conservation of angular momentum to determine the angular velocity of a rotating system in which the moment of inertia is changing
- Explain how the rotational kinetic energy changes when a system undergoes changes in both moment of inertia and angular velocity

#### Precession of a Gyroscope

- Describe the physical processes underlying the phenomenon of precession
- Calculate the precessional angular velocity of a gyroscope

#### 11.4 Conditions for Static Equilibrium

- Identify the physical conditions of static equilibrium.
- Draw a free-body diagram for a rigid body acted on by forces.
- Explain how the conditions for equilibrium allow us to solve statics problems.

#### 11.5 Examples of Static Equilibrium

- Identify and analyze static equilibrium situations
- Set up a free-body diagram for an extended object in static equilibrium
- Set up and solve static equilibrium conditions for objects in equilibrium in various physical situations

#### Stress, Strain, and Elastic Modulus

- Explain the concepts of stress and strain in describing elastic deformations of materials
- Describe the types of elastic deformation of objects and materials